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If x2+6x−5=0x^2+6x-5=0, what are the values of xx?

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Full solution

Q. If x2+6x−5=0x^2+6x-5=0, what are the values of xx?
  1. Given Quadratic Equation: We are given a quadratic equation x2+6x−5=0x^2 + 6x - 5 = 0. To find the values of xx, we can use the quadratic formula 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.
  2. Substitute Coefficients: In our equation, a=1a = 1, b=6b = 6, and c=−5c = -5. We substitute these values into the quadratic formula to find the values of xx.
  3. Calculate Discriminant: Now we calculate the discriminant, which is the part under the square root in the quadratic formula: b2−4acb^2 - 4ac. So we have 62−4(1)(−5)=36+20=566^2 - 4(1)(-5) = 36 + 20 = 56.
  4. Apply Quadratic Formula: Since the discriminant is positive, we will have two real and distinct solutions for xx. We proceed with the quadratic formula: x=−6±562×1x = \frac{-6 \pm \sqrt{56}}{2 \times 1}.
  5. Simplify Square Root: We simplify the square root of 5656 to 4×14=214\sqrt{4 \times 14} = 2\sqrt{14}. Now we have x=−6±2142x = \frac{-6 \pm 2\sqrt{14}}{2}.
  6. Divide by 22: We can simplify the expression further by dividing both terms in the numerator by 22: x=(−3±14)x = (-3 \pm \sqrt{14}).
  7. Final Solutions: Therefore, the solutions for the equation x2+6x−5=0x^2 + 6x - 5 = 0 are x=−3+14x = -3 + \sqrt{14} and x=−3−14x = -3 - \sqrt{14}.

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