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Answer:
1. For the equation x² + 5x + 4 = 0:
a. Start with the equation in the form ax² + bx + c = 0.
b. In this case, a = 1, b = 5, and c = 4.
c. To complete the square, we need to find the value that, when added and subtracted to the equation, will create a perfect square trinomial.
d. Need to rewrite the equation as a perfect square trinomial: x² + 5x + 4 = (x + a)² = x² + 2ax + a².
e. To find 'a', we have 2a = 5. => a = 5/2 = 2.5.
f. Since (x + a)² = x² + 2ax + a², we know that x² + 5x + 4 = (x + 2.5)²
g. To complete the square, add and subtract (5/2)² = 6.25.
h. The equation becomes: x² + 5x + 6.25 - 6.25 + 4 = 0.
i. Simplify: (x + 2.5)² - 2.25 = 0.
j. So, (x + 2.5)² = 2.25.
k. Taking the square root: x + 2.5 = ±√2.25.
Solving for x:
x + 2.5 = ±1.5
x = -2.5 ± 1.5
Therefore, x = -4 or x = -1.
2. For the equation 2x² - 3x - 9 = 0:
a. Start with the equation in the form ax² + bx + c = 0.
b. In this case, a = 2, b = -3, and c = -9.
c. Rewrite the equation as a perfect square trinomial: 2x² - 3x - 9 = 2(x² - 3/2x) - 9 = 2[(x - 3/4)^2 - 9/16] - 9.
d. Expand: 2(x - 3/2)^2 - 18/16 - 9 = 2(x - 3/2)^2 - 18/16 - 144/16.
e. Combine constants: 2(x - 3/2)^2 - 162/16 = 0.
f. Simplify the equation: 2(x - 3/2)^2 - 162/16 = 0.
g. Divide by 2: (x - 3/2)^2 - 81/16 = 0.
h. Add 81/16 to both sides: (x - 3/2)^2 = 81/16.
i. Taking the square root: x - 3/2 = ±√(81/16).
Solving for x:
x - 3/2 = ±(9/4)
x = 3/2 ± (9/4)
Therefore, x = -3/4 or x = 6.
Therefore, the solutions for the given quadratic equations by completing the square are:
1. x = -4 or x = -1 for the equation x² + 5x + 4 = 0.
2. x = -3/4 or x = 6 for the equation 2x² - 3x - 9 = 0.