Question

How do you solve $ - 5{x^2} + 10x + 15 = 0$ ?

Answer 1

Hint: The given problem requires us to solve a quadratic equation using a quadratic formula. There are various methods that can be employed to solve a quadratic equation like completing the square method, using quadratic formulas and by splitting the middle term. Using the quadratic formula gives us the roots of the equation directly with ease.

Complete step-by-step solution:
In the given question, we are required to solve the equation $ - 5{x^2} + 10x + 15 = 0$ with the help of quadratic formula. The quadratic formula can be employed for solving an equation only if we compare the given equation with the standard form of quadratic equation.
First of all, dividing both sides of the equation by $ - 5$, we get,
$ \Rightarrow {x^2} - 2x - 3 = 0$
Comparing with standard quadratic equation $a{x^2} + bx + c = 0$
Here,$a = 1$, $b = - 2$ and$c = - 3$.
Now, Using the quadratic formula, we get the roots of the equation as:
$x = \dfrac{{( - b) \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Substituting the values of a, b, and c in the quadratic formula, we get,
$x = \dfrac{{ - \left( { - 2} \right) \pm \sqrt {{{\left( { - 2} \right)}^2} - 4 \times 1 \times \left( { - 3} \right)} }}{{2 \times 1}}$
$ \Rightarrow x = \dfrac{{2 \pm \sqrt {4 + 12} }}{2}$
$ \Rightarrow x = \dfrac{{2 \pm \sqrt {16} }}{2}$
Now, we know that square root of $16$ is $4$,
$ \Rightarrow x = \dfrac{{2 \pm 4}}{2}$
So, $x = \dfrac{6}{2} = 3$ and $x = \left( { - 1} \right)$ are the roots of the equation $ - 5{x^2} + 10x + 15 = 0$.

Note: Quadratic equations are the polynomial equations with degree of the variable or unknown as $2$. Quadratic equations can be solved by splitting the middle term, factorizing common factors, using the quadratic formula and completing the square method. The given equation can be solved by each and every method listed above.