Answer :

Sure! Let's solve the quadratic equation step by step to find the values of [tex]\( x \)[/tex].

We are given the equation:

[tex]\[ 21x^2 + 5x - 35 = 3x^2 - 4x \][/tex]

First, we want to move all the terms to one side of the equation to set it to zero:

1. Subtract [tex]\( 3x^2 \)[/tex] from both sides:
[tex]\[
21x^2 + 5x - 35 - 3x^2 = -4x
\][/tex]

2. Add [tex]\( 4x \)[/tex] to both sides:
[tex]\[
21x^2 + 5x - 35 - 3x^2 + 4x = 0
\][/tex]

Now, simplify this equation:

- Combine the [tex]\( x^2 \)[/tex] terms: [tex]\( 21x^2 - 3x^2 = 18x^2 \)[/tex]
- Combine the [tex]\( x \)[/tex] terms: [tex]\( 5x + 4x = 9x \)[/tex]

So now the equation looks like this:

[tex]\[ 18x^2 + 9x - 35 = 0 \][/tex]

Next, we solve the quadratic equation [tex]\( 18x^2 + 9x - 35 = 0 \)[/tex] by using the quadratic formula:

The quadratic formula is:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

For our equation, [tex]\( a = 18 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = -35 \)[/tex].

Plug these values into the quadratic formula:

[tex]\[ x = \frac{-9 \pm \sqrt{9^2 - 4 \cdot 18 \cdot (-35)}}{2 \cdot 18} \][/tex]

1. Calculate [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
9^2 = 81
\][/tex]
[tex]\[
-4 \cdot 18 \cdot (-35) = 4 \cdot 18 \cdot 35 = 2520
\][/tex]
[tex]\[
b^2 - 4ac = 81 + 2520 = 2601
\][/tex]

2. Find the square root of 2601:
[tex]\[
\sqrt{2601} = 51
\][/tex]

3. Substitute back into the quadratic formula:
[tex]\[
x = \frac{-9 \pm 51}{36}
\][/tex]

This gives us two potential solutions:

1. [tex]\( x = \frac{-9 + 51}{36} = \frac{42}{36} = \frac{7}{6} \)[/tex]

2. [tex]\( x = \frac{-9 - 51}{36} = \frac{-60}{36} = \frac{-5}{3} \)[/tex]

Therefore, the solutions to the equation are:

[tex]\[ x = \frac{7}{6} \quad \text{and} \quad x = \frac{-5}{3} \][/tex]

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