Basic Quadratic Equation Questions
Quadratic equations are an important part of algebra, and as students, we must all be familiar with their definition and the ways of solving quadratic equation problems. In this article, we are going to familiarize the students with all the concepts surrounding quadratic equations and the methods of solving problems related to this topic. A short definition of a quadratic equation would be: a quadratic equation is a second-degree polynomial, which we represent as ‘ax2+ bx + c’ in general.
In this representation, a cannot be equal to 0 and b,c are known as coefficients and are constant by nature. With this basic introduction, let's move forward with a formal definition, formulae and detailed solutions to quadratic equation questions to enable better understanding.
Why are Quadratic Equations Important?
Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and even computer science. They help us model real-world scenarios like projectile motion, population growth, and electrical circuit analysis. Understanding quadratics is crucial for success in higher-level math and science courses.
Definition of Quadratic Equations
A quadratic equation is a polynomial where the highest power of the variable is 2. We generally represent it as ax2 + bx + c. Here a, b and c are real numbers or constants, and x is the variable. In this case, the value of a cannot be 0 as that would remove the x2 term, and the equation won't be quadratic after that.
A quadratic equation is an equation of second degree with more than two terms. It means that at least one of the terms of the equation is squared. In the above-given equation. The answer to the equation also known as the roots of the equation is the value of the “x”. The value of the “x” has to satisfy the equation.
Some examples of quadratic equations can be as follows:
56x2 + ⅔ x + 1, where a = 56, b = ⅔ and c = 1.
-4/3 x2 + 64x - 30, where a = -4/3, b = 64 and c = -30.
Roots of a Quadratic Equation
To solve basic quadratic equation questions or any quadratic equation problems, we need to solve the equation. Solving quadratic equations gives us the roots of the polynomial. The roots of the equation are the values of x at which ax2 + bx + c = 0. Since a quadratic equation is a polynomial of degree 2, we obtain two roots in this case.
There are several methods for solving quadratic equation problems, as we can see below:
Factorization Method.
Completing The Square Method.
Quadratic Equation Formula.
Quadratic Equation Formula
So what is the quadratic equation formula? The quadratic equation formula or the Sridharacharya Formula is a method for finding out the roots of two-degree polynomials. This formula helps solve quadratic equation problems. The formula is as given below:
\[ x = \frac{-b \pm\sqrt{b^{2}-4ac}}{2a} \]
Where x represents the roots of the equation and (b2−4ac) is the discriminant.
By finding out the value of the discriminant, we can predict the nature of the roots. There are three possibilities with three different implications:
Two distinct roots which are real, if b2 - 4ac > 0.
Two real roots equal in magnitude, if b2 - 4ac = 0.
Imaginary roots or absence of real roots if b2 - 4ac < 0.
Now that the basic principles of quadratic equations are clear, we will move on to some solved examples. But before that, let us list some quadratic equation questions for the students to solve.
Quadratic Equation Practice Questions
The following are a list of questions for you to solve once you have gone through the quadratic equation questions and answers in the solved examples section:
Find the determinant of the following quadratic equations: 2x2 + 3x + 6, 70x2 + 49 + 14, ⅔ y2 + 63y + 42.
Find the roots of the following quadratic equations: x2 - 45x + 324, 2x2 - 22x + 42, ½ x2 + 2x + 4.
The product of two consecutive numbers is 420, and their sum is 41. Find the numbers.
Before solving these, let's check out the solved examples with questions and answers on the quadratic equation.
Solved Examples
1. Solve x2 + 5x + 6 = 20 by factorization method.
Solution: The given polynomial or quadratic equation is
x2 + 5x + 6 = 20
Solving by factorization method,
x2 + 5x + 6 - 20 = 0.
or, x2 + 5x - 14 = 0
or, x2 - 2x + 7x - 14 = 0
or, x(x - 2) +7(x - 2) = 0
or, (x - 2)(x + 7) = 0
or, (x - 2) = 0, (x + 7) = 0
or, x = +2, -7.
2. Solve 2x2 - 5x + 3 using the quadratic equation formula.
Solution: The quadratic equation formula is:
\[ x = \frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \]
The determinant or b2-4ac = (-5)2 - 4 × 3 × 2 = 25 - 24 = 1
\[ \sqrt{b^{2} - 4ac} = 1 \]
Therefore, \[ x= \frac{-(-5)\pm1}{2\times2} \]
\[ x = \frac{5+1}{4} = \frac{6}{4} = \frac{3}{2} \]
\[ x = \frac{5-1}{4} = \frac{4}{4} = 1 \]
Thus the roots of the equation are 3/2 and 1.
3. Solve the quadratic equation: \[x^2 - 4x + 4 = 0\]
Solution: Given quadratic equation is a perfect square trinomial, so we can factor it directly.
\[(x - 2)^2 = 0\]
Taking the square root of both sides:
\[x - 2 = 0\]
Solving for \(x\):
\[x = 2\]
4. Solve the quadratic equation: \[2x^2 - 5x + 2 = 0\]
Solution: Using the quadratic formula:
\[x = \dfrac{5 \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)}\]
\[x = \dfrac{5 \pm \sqrt{25 - 16}}{4}\]
\[x = \dfrac{5 \pm \sqrt{9}}{4}\]
\[x = \dfrac{5 \pm 3}{4}\]
Two solutions:
\[x = \dfrac{8}{4} = 2\]
\[x = \dfrac{2}{4} = \dfrac{1}{2}\]
5. Solve the quadratic equation: \[3x^2 + 7x + 2 = 0\]
Solution: Using the quadratic formula:
\[x = \dfrac{-7 \pm \sqrt{7^2 - 4(3)(2)}}{2(3)}\]
\[x = \dfrac{-7 \pm \sqrt{49 - 24}}{6}\]
\[x = \dfrac{-7 \pm \sqrt{25}}{6}\]
\[x = \dfrac{-7 \pm 5}{6}\]
Two solutions:
\[x = \dfrac{-2}{6} = -\dfrac{1}{3}\]
\[x = \dfrac{-12}{6} = -2\]
6. Solve the quadratic equation: \[4x^2 + 8x + 2 = 0\]
Solution: Divide the equation by the common factor (in this case, 2):
\[2x^2 + 4x + 1 = 0\]
Using the quadratic formula:
\[x = \dfrac{-4 \pm \sqrt{4^2 - 4(2)(1)}}{2(2)}\]
\[x = \dfrac{-4 \pm \sqrt{16 - 8}}{4}\]
\[x = \dfrac{-4 \pm \sqrt{8}}{4}\]
\[x = \dfrac{-4 \pm 2\sqrt{2}}{4}\]
Simplify:
\[x = \dfrac{-2 \pm \sqrt{2}}{2}\]
Two solutions:
\[x = \dfrac{-2 + \sqrt{2}}{2}\]
\[x = \dfrac{-2 - \sqrt{2}}{2}\]
7. Solve the quadratic equation by factorization method: \[x^2 - 5x + 6 = 0\]
Solution: Factor the quadratic equation:
\[(x - 2)(x - 3) = 0\]
Setting each factor to zero:
\[x - 2 = 0 \quad \text{or} \quad x - 3 = 0\]
Solving for \(x\):
\[x = 2 \quad \text{or} \quad x = 3\]
8. Solve the quadratic equation: \[6x^2 - 11x + 4 = 0\]
Solution: Using the quadratic formula:
\[x = \dfrac{11 \pm \sqrt{(-11)^2 - 4(6)(4)}}{2(6)}\]
\[x = \dfrac{11 \pm \sqrt{121 - 96}}{12}\]
\[x = \dfrac{11 \pm \sqrt{25}}{12}\]
\[x = \dfrac{11 \pm 5}{12}\]
Two solutions:
\[x = \dfrac{16}{12} = \dfrac{4}{3}\]
\[x = \dfrac{6}{12} = \dfrac{1}{2}\]
9. Solve the quadratic equation: \[2x^2 + 3x - 5 = 0\]
Solution: Using the quadratic formula:
\[x = \dfrac{-3 \pm \sqrt{3^2 - 4(2)(-5)}}{2(2)}\]
\[x = \dfrac{-3 \pm \sqrt{9 + 40}}{4}\]
\[x = \dfrac{-3 \pm \sqrt{49}}{4}\]
\[x = \dfrac{-3 \pm 7}{4}\]
Two solutions:
\[x = \dfrac{4}{4} = 1\]
\[x = \dfrac{-10}{4} = -\dfrac{5}{2}\]
10. Solve the quadratic equation: \[5x^2 - 4x - 3 = 0\]
Solution: Using the quadratic formula:
\[x = \dfrac{4 \pm \sqrt{(-4)^2 - 4(5)(-3)}}{2(5)}\]
\[x = \dfrac{4 \pm \sqrt{16 + 60}}{10}\]
\[x = \dfrac{4 \pm \sqrt{76}}{10}\]
\[x = \dfrac{4 \pm 2\sqrt{19}}{10}\]
Simplify:
\[x = \dfrac{2 \pm \sqrt{19}}{5}\]
Two solutions:
\[x = \dfrac{2 + \sqrt{19}}{5}\]
\[x = \dfrac{2 - \sqrt{19}}{5}\]
This is all about the roots of quadratic equations and their formulas. Learn the formulas and find out how they are used to derive the roots of an equation easily.
Conclusion
By diving into this chapter, you'll build a solid understanding of quadratic equations. Mastering these equations will empower you to effortlessly solve any quadratic problem, use your skills in real-life situations, and amaze your teachers with your math abilities. Keep in mind that practice is crucial! Take on the practice questions and reach out for assistance if necessary. The realm of math is ready for you, and quadratic equations are your ticket to exploring its intriguing complexities!
FAQs on Quadratic Equation Questions
1. Define a quadratic equation along with suitable examples. also, state the quadratic equation formula.
A quadratic equation is a polynomial where the highest power of the variable is neither more nor less than 2. So essentially, a quadratic equation is a polynomial of degree 2. We represent such an equation in a general format as ax2 + bx + c, where a, b and c are known as the coefficients or the constants of the equation. The thumb rule for quadratic equations is that the value of a cannot be 0. The x in the expression is the variable. This algebraic expression, when solved, will yield two roots.
Some examples of quadratic equations are:
3x2 + 4x + 7 = 34
x2 + 8x + 12 = 40
The quadratic equation formula is a method for solving quadratic equation questions. The formula is as follows:
\[ x = \frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \]
where x represents the roots of the equation.
2. What are the roots of a quadratic equation? What are the zeroes of a polynomial?
The roots of a quadratic equation are the values obtained when we solve the equation. They are those values of x for which the expression ax2+bx+c becomes equal to 0. These values are also known as the zeroes of the polynomial. Since a quadratic equation is essentially a polynomial of degree 2, we get two roots after solving the given polynomial.
As we practice more and more quadratic equation sums, our ideas regarding which method to use while solving a given question will get clearer.
3. How can I solve a quadratic equation?
Quadratic equation is an equation with more than one term in it and at least one of the terms having degree 2. Its general form is ax2 + bx + c, whereas a,b,c are real numbers and a is not equal to zero. The values which satisfy the “x” in the equation are the solution for the quadratic equation. They are also known as the roots of the equation. The quadratic equation can be solved in the algebraic method and graphical method.
In the algebraic method, the equation is reduced to the roots by shifting terms from L.H.S to the R.H.S and using different mathematical operations. In the graphic method, the equation is solved by drawing it on the map and solving it using the parabola the equation makes on the graph. The value of “a” determines whether the graph of the equation is concave parabola or convex parabola. The value of the discriminant decides whether the curve will intersect the x-axis or not.
4. What are the applications of quadratic equations?
A quadratic equation is a polynomial equation with more than one term and at least one of the terms having 2 as square. It is generally used in different situations in day-to-day life. In constructing rooms and boxes of different geometric shapes. If you want to construct a box made of wood with 5 square feet dimensions, you can write a quadratic equation to measure its area and calculate the material required.
It can also be used in selling something and calculating the profit and loss you may incur after selling the good. To know it, you can simply form a quadratic equation. It can also be used in athletics while throwing objects like a javelin, shot put ball, etc. It can also be used to calculate the distance. Generally, when someone travels up and down the river uses this equation to measure the distance to be travelled.
5. How are quadratic equations used in athletics and construction?
Quadratic equations are equations with at least one term having 2 as a square and it has more than one term in the equation preferably, four terms. The general form of the equation is ax2 + bx+c whereas a,b,c are real numbers and a is not equal to zero. The values which satisfy the “x” in the equation are the solution for the quadratic equation. They are also known as the roots of the equation. In exams, they have preferable weightage and practising them correctly will improve the marks. On the other hand, they are also used in real-life situations. In athletics, it is used to measure the speed and force to be applied to throw an object like an arrow, shot put the ball, discus, etc. They use the velocity equation to measure the height of the ball from which it should be thrown. In the field of construction, a quadratic equation is framed with the known dimensions of the building or a room to figure out unknown values like the area, perimeter, etc.