Question

Solve the following equation using the quadratic equation formula $4{x^2} - 3x - 7 = 0$.

Answer 1

Hint: In the given question we are given a quadratic equation of type \[a{x^2} + bx + c = 0\]. For solving this form of the quadratic equation we will use the quadratic equation formula which is, \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. We will identify the values of a,b, and c from the given equation and then solve the equation using the formula.

Complete step by step solution:
There are several ways of solving a quadratic equation without using the quadratic equation formula they are, factorizing, completing the square method, graphing, etc.
All these methods are sometimes complex to use, one of the most sure-shot methods of solving an equation is by using the quadratic equation formula.
A general equation of the form \[a{x^2} + bx + c = 0\], where x is the unknown and a,b and c are the constants with \[a \ne 0\].
The quadratic formula is
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
The plus-minus sign\[ \pm \]in the middle of the formula represents the fact that the equation has two solutions.
When written separately we get,
\[{x_1} = \dfrac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}}\]
And
\[{x_2} = \dfrac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}\]

In order to use the quadratic equation formula, it is imperative that the equation should be arranged in the form “quadratic=0”.
Now, given equation
$4{x^2} - 3x - 7 = 0$
We know that our given equation is of the standard form,
Comparing the coefficient of $4{x^2} - 3x - 7 = 0$ with, we have, \[a = 4\],\[b = - 3\] and \[c = - 7\].
Substituting the values in \[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\],
We get,
\[x = \dfrac{{3 \pm \sqrt {{{\left( { - 3} \right)}^2} - 4\left( 4 \right)\left( { - 7} \right)} }}{{2\left( 4 \right)}}\]
Thus,
\[x = 3 \pm \dfrac{{\sqrt {9 + 112} }}{8}\]
\[x = 3 \pm \dfrac{{\sqrt {121} }}{8}\]
Therefore,
\[{x_1} = 3 + \dfrac{{\sqrt {121} }}{8}\]
\[{x_2} = 3 - \dfrac{{\sqrt {121} }}{8}\],
Hence,
\[{x_1} = 3 + \dfrac{{11}}{8}\] Or \[\dfrac{{35}}{8}\]
\[{x_2} = 3 - \dfrac{{11}}{8}\] Or \[\dfrac{{13}}{8}\]

Note: One of the easier ways to solve the equation using the formula is to identify the values of a,b, and c at first. Putting the values directly into the formula increases the chance of committing mistakes related to signs. Remember “\[{b^2}\]” means square of all of \[b\] including its sign.