Question

is the solution shown below correct? explain.

(3x + 2 = 8x^2 + 6x)

(-8x^2 - 3x + 2 = 0)

(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(-8)(2)}}{2(-8)})

(x = \frac{3 \pm \sqrt{9 - (-64)}}{-16})

(x = \frac{3 \pm \sqrt{73}}{-16})

note: the image shows a students incorrect work with errors in signs and calculations:
(3x + 2 = 8x^2 + 6x)

(-8x^2 + 3x + 2 = 0)

(x = \frac{-3 \pm \sqrt{(3)^2 - (4)(-8)(2)}}{-16})

(x = \frac{-3 \pm \sqrt{9 - (64)}}{-16})

(x = \frac{3 \pm \sqrt{55}i}{16})

Explanation:

⚡ Using what you learned: quadratic formula and its applications

Step 1: Analyze the initial equation setup

The original equation is:

$$ 9x + 2 = 8x^2 + 6x $$

Rearranging all terms to one side to set the quadratic equation to \( 0 \):

$$ -8x^2 + (9x - 6x) + 2 = 0 $$

$$ -8x^2 + 3x + 2 = 0 $$

This step is correct. The coefficients are:

$$ a = -8, \quad b = 3, \quad c = 2 $$

Step 2: Analyze the quadratic formula substitution

The quadratic formula is:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Substituting the coefficients:

$$ x = \frac{-(3) \pm \sqrt{(3)^2 - 4(-8)(2)}}{2(-8)} $$

$$ x = \frac{-3 \pm \sqrt{(3)^2 - 4(-8)(2)}}{-16} $$

This substitution step shown in the image is correct.

Step 3: Analyze the simplification of the discriminant

Now evaluate the terms inside the square root (the discriminant):

$$ (3)^2 = 9 $$

$$ -4(-8)(2) = -(-64) = +64 $$

So the expression under the radical should be:

$$ 9 - (-64) = 9 + 64 = 73 $$

In the image, the third line shows:

$$ x = \frac{-3 \pm \sqrt{9 - (64)}}{-16} $$

The error occurs here. The term \(-4(-8)(2)\) equals \(+64\), which means we add \(64\) to \(9\). The image incorrectly subtracts \(64\) from \(9\), writing \(9 - (64)\) instead of \(9 - (-64)\) or \(9 + 64\).

Step 4: Analyze the final simplification step

Because of the subtraction error, the image calculates:

$$ 9 - 64 = -55 $$

$$ \sqrt{-55} = \sqrt{55}i $$

This leads to the incorrect final line:

$$ x = \frac{3 \pm \sqrt{55}i}{16} $$

The correct calculation should be:

$$ x = \frac{-3 \pm \sqrt{73}}{-16} = \frac{3 \mp \sqrt{73}}{16} $$

Answer:

No, the solution is not correct.

The error occurs in the third line when simplifying the expression under the radical (the discriminant). The term \(-4ac\) is \(-4(-8)(2)\), which equals \(+64\). Therefore, the expression under the radical should be \(9 + 64 = 73\).

Instead, the work incorrectly subtracts \(64\) from \(9\), resulting in \(\sqrt{-55}\) and introducing an imaginary unit \(i\). The correct solution is:

$$ x = \frac{3 \pm \sqrt{73}}{16} $$