Question

How do you solve $ - 3{x^2} - 5 = 22$ ?

Answer 1

Hint:This equation is the quadratic equation. The general form of the quadratic equation is $a{x^2} + bx + c = 0$. Where ‘a’ is the coefficient of ${x^2}$, ‘b’ is the coefficient of x and ‘c’ is the constant term.
-To solve this equation, we will apply the quadratic formula for the quadratic equation.
The quadratic formula is as below:
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ .
-Here, $\sqrt {{b^2} - 4ac} $ is called the discriminant. And it is denoted by $\Delta $.
-If $\Delta $ is greater than 0, then we will get two distinct and real roots.
-If $\Delta $ is less than 0, then we will not get real roots. In this case, we will get two complex numbers.
-If $\Delta $ is equal to 0, then we will get two equal real roots.

Complete step by step answer:Here, the given quadratic equation is
$ \Rightarrow - 3{x^2} - 5 = 22$
Let us subtract 22 on both sides.
$ \Rightarrow - 3{x^2} - 5 - 22 = 22 - 22$
That is equal to,
$ \Rightarrow - 3{x^2} - 27 = 0$
Let us take out -3 as a common factor.
$ \Rightarrow - 3\left( {{x^2} + 9} \right) = 0$
That is equal to,
$ \Rightarrow {x^2} + 9 = 0$
We want to find the roots.
First, let us compare the above expression with $a{x^2} + bx + c = 0$.
Here, we get the value of ‘a’ is 1, the value of ‘b’ is 0, and the value of ‘c’ is 9.
Now, let us find the discriminant$\Delta $.
$ \Rightarrow \Delta = {b^2} - 4ac$
Let us substitute the values.
$ \Rightarrow \Delta = {\left( 0 \right)^2} - 4\left( 1 \right)\left( 9 \right)$
Simplify it.
$ \Rightarrow \Delta = 0 - 36$
Subtract the right-hand side.
$ \Rightarrow \Delta = - 36$
Here, $\Delta $ is less than 0, then we cannot get real roots.

Note:
One important thing is, we can always check our work by multiplying out factors back together, and check that we have got back the original answer.
Here is a list of methods to solve quadratic equations:
-Factorization
-Completing the square
-Using graph
-Quadratic formula