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Question: How do you solve \(3{x^2} + 21x = - 36\) using the quadratic formula?...

How do you solve 3x2+21x=−363{x^2} + 21x = - 36 using the quadratic formula?

Explanation

Solution

Given a quadratic equation. We have to find the solution of the equation by applying the quadratic formula. First, we will apply the formula to solve the quadratic equation. Then, apply the basic arithmetic operation to the expression. Then, write the solution of the equation.
Formula used:
The quadratic formula to solve the quadratic equation of the form ax2+bx+c=0a{x^2} + bx + c = 0 is given by:
x=−b±b2−4ac2ax = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}

Complete step by step solution:
We are given a quadratic equation, 3x2+21x=−363{x^2} + 21x = - 36.
First, we will write the equation in the standard form of the quadratic equation by adding 3636on both sides of equation.
⇒3x2+21x+36=−36+36\Rightarrow 3{x^2} + 21x + 36 = - 36 + 36
⇒3x2+21x+36=0\Rightarrow 3{x^2} + 21x + 36 = 0
Now, we will find the value of a, b and c by comparing the equation with the standard form of quadratic equation, ax2+bx+c=0a{x^2} + bx + c = 0.
⇒a=3\Rightarrow a = 3, b=21b = 21 and c=36c = 36
Then, we will apply the quadratic formula.
⇒x=−21±(21)2−4×3×362×3\Rightarrow x = \dfrac{{ - 21 \pm \sqrt {{{\left( {21} \right)}^2} - 4 \times 3 \times 36} }}{{2 \times 3}}
On simplifying the expression, we get:
⇒x=−21±441−4326\Rightarrow x = \dfrac{{ - 21 \pm \sqrt {441 - 432} }}{6}
⇒x=−21±96\Rightarrow x = \dfrac{{ - 21 \pm \sqrt 9 }}{6}
Further simplify the expression.
⇒x=−21±36\Rightarrow x = \dfrac{{ - 21 \pm 3}}{6}
Now, we will find the solution of the equation by solving the expression.
⇒x=−21+36 or x=−21−36\Rightarrow x = \dfrac{{ - 21 + 3}}{6}{\text{ or }}x = \dfrac{{ - 21 - 3}}{6}
Further simplify the expression.
⇒x=−186 or x=−246\Rightarrow x = \dfrac{{ - 18}}{6}{\text{ or }}x = \dfrac{{ - 24}}{6}
On simplifying the expression, we get:
⇒x=−3 or x=−4\Rightarrow x = - 3{\text{ or }}x = - 4
Hence, the solution of the expression x=−3x = - 3 and x=−4x = - 4

Note: Please note that the roots of the quadratic equation depend on the value of the discriminant of the equation. The discriminant of the equation is calculated as D=b2−4acD = {b^2} - 4ac of the equation of the form ax2+bx+c=0a{x^2} + bx + c = 0. If the discriminant of the equation is less than zero then the roots of the equation must be imaginary. If the discriminant of the equation is equal to zero then the equation has 1 real root. If the discriminant of the equation is greater than zero then the equation has 2 real roots.