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Solve using the quadratic formula.\newlinex2−5x−5=0x^2 - 5x - 5 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinex=x = _____ or x=x = _____

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Q. Solve using the quadratic formula.\newlinex2−5x−5=0x^2 - 5x - 5 = 0\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinex=x = _____ or x=x = _____
  1. Quadratic Formula: The quadratic formula is given by x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are the coefficients of the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0. In this case, a=1a = 1, b=−5b = -5, and c=−5c = -5.
  2. Calculate Discriminant: First, calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac. For our equation, the discriminant is (−5)2−4(1)(−5)=25+20=45(-5)^2 - 4(1)(-5) = 25 + 20 = 45.
  3. Apply Quadratic Formula: Now, apply the quadratic formula with the values of aa, bb, and cc. We have two solutions, one for the addition and one for the subtraction:\newlinex=−(−5)±452×1x = \frac{-(-5) \pm \sqrt{45}}{2 \times 1}\newlinex=5±452x = \frac{5 \pm \sqrt{45}}{2}
  4. Simplify Square Root: Simplify the square root of 4545. Since 45=9×545 = 9 \times 5 and 9=3\sqrt{9} = 3, we can write 45\sqrt{45} as 353\sqrt{5}. So the solutions become:\newlinex=5±352x = \frac{5 \pm 3\sqrt{5}}{2}
  5. Approximate Square Root: Now we have two solutions for xx:x=5+352x = \frac{5 + 3\sqrt{5}}{2}x=5−352x = \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 5\sqrt{5}.
  6. Perform Calculations: Approximate 5\sqrt{5} using a calculator. 5\sqrt{5} is approximately 2.2362.236. Now substitute this approximation into the solutions:\newlinex≈(5+3×2.236)/2x \approx (5 + 3 \times 2.236) / 2\newlinex≈(5−3×2.236)/2x \approx (5 - 3 \times 2.236) / 2
  7. Divide by 22: Perform the calculations:\newlinex≈(5+6.708)/2x \approx (5 + 6.708) / 2\newlinex≈(5−6.708)/2x \approx (5 - 6.708) / 2\newlinex≈11.708/2x \approx 11.708 / 2\newlinex≈−1.708/2x \approx -1.708 / 2
  8. Divide by 22: Perform the calculations:\newlinex≈(5+6.708)/2x \approx (5 + 6.708) / 2\newlinex≈(5−6.708)/2x \approx (5 - 6.708) / 2\newlinex≈11.708/2x \approx 11.708 / 2\newlinex≈−1.708/2x \approx -1.708 / 2Finally, divide by 22 to get the approximate decimal solutions:\newlinex≈11.708/2≈5.854x \approx 11.708 / 2 \approx 5.854\newlinex≈−1.708/2≈−0.854x \approx -1.708 / 2 \approx -0.854\newlineRound these to the nearest hundredth:\newlinex≈5.85x \approx 5.85\newlinex≈−0.85x \approx -0.85

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