Solve the given quadratic equation exactly using the quadratic formula. Write the solutions in its simplest form. Using a calculator, determine all irrational solutions to the nearest thousandth. Separate the solutions with a comma and list the smaller one first.
For the quadratic equation (x - 10 )(x + 9 ) = 2, the solution(s) is(are)
To use the quadratic formula, you need to arrange things so that you have
y = ax^2 + bx + c
So,
(x - 10 )(x + 9 ) = 2
x^2-x-90 = 2
x^2 - x - 92 = 0
Now just apply the formula.
To solve the quadratic equation (x - 10)(x + 9) = 2, we need to find the values of x that satisfy this equation.
Step 1: Expand the equation
(x - 10)(x + 9) = 2
x^2 + 9x - 10x - 90 = 2
x^2 - x - 90 = 2
Step 2: Move all the terms to one side, so the equation becomes equal to zero.
x^2 - x - 90 - 2 = 0
x^2 - x - 92 = 0
Step 3: Identify the values of a, b, and c in the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -1, and c = -92.
Step 4: Substitute the values of a, b, and c into the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4(1)(-92))) / (2(1))
x = (1 ± √(1 + 368)) / 2
x = (1 ± √(369)) / 2
Step 5: Simplify the expression (√(369)) to its simplest form.
√(369) cannot be simplified further because it is not a perfect square.
Step 6: Calculate the approximate value of (√(369)) using a calculator.
√(369) is approximately 19.209.
Step 7: Evaluate the two solutions for x using the quadratic formula.
x = (1 + 19.209) / 2 ≈ 10.604
x = (1 - 19.209) / 2 ≈ -9.604
Therefore, the exact solutions to the quadratic equation (x - 10)(x + 9) = 2 are x ≈ 10.604 and x ≈ -9.604.