Solve using the quadratic formula. $\newline$ $x^2 - 5x - 5 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $x =$ _ or $x =$ _

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Solve a quadratic equation using the quadratic formula

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Q. Solve using the quadratic formula.

\newline

x^2 - 5x - 5 = 0

\newline

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

\newline

x =

_____ or

x =

_____

Quadratic Formula: The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$ . In this case, $a = 1$ , $b = -5$ , and $c = -5$ .
Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ . For our equation, the discriminant is $(-5)^2 - 4(1)(-5) = 25 + 20 = 45$ .
Apply Quadratic Formula: Now, apply the quadratic formula with the values of $a$ , $b$ , and $c$ . We have two solutions, one for the addition and one for the subtraction: $\newline$ $x = \frac{-(-5) \pm \sqrt{45}}{2 \times 1}$ $\newline$ $x = \frac{5 \pm \sqrt{45}}{2}$
Simplify Square Root: Simplify the square root of $45$ . Since $45 = 9 \times 5$ and $\sqrt{9} = 3$ , we can write $\sqrt{45}$ as $3\sqrt{5}$ . So the solutions become: $\newline$ $x = \frac{5 \pm 3\sqrt{5}}{2}$
Approximate Square Root: Now we have two solutions for $x$ : $x = \frac{5 + 3\sqrt{5}}{2}$ $x = \frac{5 - 3\sqrt{5}}{2}$ These are the exact solutions in terms of square roots. To express them as decimals rounded to the nearest hundredth, we need to approximate $\sqrt{5}$ .
Perform Calculations: Approximate $\sqrt{5}$ using a calculator. $\sqrt{5}$ is approximately $2.236$ . Now substitute this approximation into the solutions: $\newline$ $x \approx (5 + 3 \times 2.236) / 2$ $\newline$ $x \approx (5 - 3 \times 2.236) / 2$
Divide by $2$ : Perform the calculations: $\newline$ $x \approx (5 + 6.708) / 2$ $\newline$ $x \approx (5 - 6.708) / 2$ $\newline$ $x \approx 11.708 / 2$ $\newline$ $x \approx -1.708 / 2$
Divide by $2$ : Perform the calculations: $\newline$ $x \approx (5 + 6.708) / 2$ $\newline$ $x \approx (5 - 6.708) / 2$ $\newline$ $x \approx 11.708 / 2$ $\newline$ $x \approx -1.708 / 2$ Finally, divide by $2$ to get the approximate decimal solutions: $\newline$ $x \approx 11.708 / 2 \approx 5.854$ $\newline$ $x \approx -1.708 / 2 \approx -0.854$ $\newline$ Round these to the nearest hundredth: $\newline$ $x \approx 5.85$ $\newline$ $x \approx -0.85$