Find the solutions of the quadratic equation $2 x^{2}-8 x-9=0$ . $\newline$ Choose $1$ answer: $\newline$ (A) $2 \pm \frac{\sqrt{34}}{2} i$ $\newline$ (B) $-2 \pm \frac{\sqrt{34}}{2} i$ $\newline$ (C) $2 \pm \frac{\sqrt{34}}{2}$ $\newline$ (D) $1 \pm \frac{\sqrt{34}}{4}$

Full solution

Q. Find the solutions of the quadratic equation

2 x^{2}-8 x-9=0

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Choose

1

answer:

\newline

(A)

2 \pm \frac{\sqrt{34}}{2} i

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(B)

-2 \pm \frac{\sqrt{34}}{2} i

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(C)

2 \pm \frac{\sqrt{34}}{2}

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(D)

1 \pm \frac{\sqrt{34}}{4}

Identify coefficients: Identify the coefficients of the quadratic equation $2x^2 - 8x - 9 = 0$ . $\newline$ The standard form of a quadratic equation is $ax^2 + bx + c = 0$ . Here, $a = 2$ , $b = -8$ , and $c = -9$ .
Use quadratic formula: Use the quadratic formula to solve for x. The quadratic formula is $x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$ .
Calculate discriminant: Calculate the discriminant $D = b^2 - 4ac$ . $\newline$ $D = (-8)^2 - 4(2)(-9)$ $\newline$ $D = 64 + 72$ $\newline$ $D = 136$ $\newline$ The discriminant is positive, which means there are two real and distinct solutions.
Substitute values into formula: Substitute the values of $a$ , $b$ , and $D$ into the quadratic formula. $\newline$ $x = \frac{-(-8) \pm \sqrt{136}}{2 \cdot 2}$ $\newline$ $x = \frac{8 \pm \sqrt{136}}{4}$
Simplify square root: Simplify the square root of the discriminant. $\sqrt{136}$ can be simplified to $\sqrt{4 \times 34}$ , which is $2 \times \sqrt{34}$ .
Substitute simplified square root: Substitute the simplified square root back into the equation. $x = \frac{8 \pm 2 \cdot \sqrt{34}}{4}$
Simplify equation: Simplify the equation by dividing both terms in the numerator by $4$ . $\newline$ $x = \frac{8}{4} \pm \frac{2 \cdot \sqrt{34}}{4}$ $\newline$ $x = 2 \pm \frac{\sqrt{34}}{2}$
Write final solutions: Write the final solutions. $\newline$ The solutions are $x = 2 + \left(\frac{\sqrt{34}}{2}\right)$ and $x = 2 - \left(\frac{\sqrt{34}}{2}\right)$ .