Question

-9x + 10x² + 8 = 14
choose 1 answer:
a ( x = \frac{3}{4}, 1 )
b ( x = \frac{4 pm sqrt{26}}{10} )
c ( x = \frac{-1 pm sqrt{109}}{18} )
d ( x = \frac{9 pm sqrt{321}}{20} )

Explanation:

Step1: Rearrange the equation

First, we need to rewrite the given equation \(-9x + 10x^2 + 8 = 14\) in standard quadratic form \(ax^2 + bx + c = 0\). Subtract 14 from both sides:

$$ 10x^2 - 9x + 8 - 14 = 0 $$

Simplify the constants:

$$ 10x^2 - 9x - 6 = 0 $$

Wait, no, wait. Wait, the original equation is \(-9x + 10x^2 + 8 = 14\). Let's do it again. Subtract 14 from both sides: \(10x^2 - 9x + 8 - 14 = 0\) → \(10x^2 - 9x - 6 = 0\)? Wait, no, 8 - 14 is -6. But maybe I made a mistake. Wait, let's check the quadratic formula. The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) for \(ax^2 + bx + c = 0\). Wait, maybe I rearranged wrong. Let's start over.

Original equation: \(-9x + 10x^2 + 8 = 14\)

Let's write it as \(10x^2 - 9x + 8 - 14 = 0\) → \(10x^2 - 9x - 6 = 0\). Wait, but the options don't match this. Wait, maybe I misread the original equation. Wait, the original equation is \(-9x + 10x^2 + 8 = 14\)? Wait, maybe it's \(-9x + 10x^2 + 8 = 14\), or is it \(-9x + 10x^2 + 8 = 14\)? Wait, let's check the options. Option D is \(x = \frac{9 \pm \sqrt{321}}{20}\). Let's compute the discriminant for \(10x^2 - 9x - 6 = 0\): \(b^2 - 4ac = (-9)^2 - 410(-6) = 81 + 240 = 321\). Ah! There we go. So \(a = 10\), \(b = -9\), \(c = -6\). Wait, but in the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). So \( -b = 9\), so \(x = \frac{9 \pm \sqrt{(-9)^2 - 410(-6)}}{2*10}\) → \(x = \frac{9 \pm \sqrt{81 + 240}}{20}\) → \(x = \frac{9 \pm \sqrt{321}}{20}\), which is option D. Wait, but let's check the options again. Option D is \(x = \frac{9 \pm \sqrt{321}}{20}\). So that's the correct one. Wait, but let's verify.

Wait, let's re-express the original equation. Let's write it as \(10x^2 - 9x + 8 - 14 = 0\) → \(10x^2 - 9x - 6 = 0\). So \(a = 10\), \(b = -9\), \(c = -6\). Then discriminant \(D = b^2 - 4ac = (-9)^2 - 410(-6) = 81 + 240 = 321\). Then \(x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{321}}{20}\), which is option D.

Wait, but let's check the other options. Option A: \(x = 3/4, 1\). Let's plug x=1 into original equation: left side: -9(1) + 10(1)^2 + 8 = -9 + 10 + 8 = 9 ≠14. So A is wrong. Option B: \(x = (4 ±√26)/10\). Let's compute discriminant for a=10, b=-4, c=... Wait, no, if x=(4±√26)/10, then the quadratic would be 10x^2 -4x + (16 -26)/10? No, that doesn't match. Option C: (-1 ±√109)/18. Discriminant would be 1 + 418... No, not matching. So the correct answer is D.

Step2: Apply Quadratic Formula

For the quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by:

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

From the equation \(10x^2 - 9x - 6 = 0\), we identify \(a = 10\), \(b = -9\), and \(c = -6\).

Substitute these values into the quadratic formula:

$$ x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(10)(-6)}}{2(10)} $$

Simplify the numerator and denominator:

$$ x = \frac{9 \pm \sqrt{81 + 240}}{20} $$

Calculate the discriminant:

$$ 81 + 240 = 321 $$

Thus, the solutions are:

$$ x = \frac{9 \pm \sqrt{321}}{20} $$

Answer:

D. \(x = \frac{9 \pm \sqrt{321}}{20}\)