45. Express each equation in the specified form.
a. y = (x – 3)² - 25 in standard form
B. y = 2(x – 7)(x + 3) in standard form.
C. y = - 2x² + 28x – 26 in factored form
D. y = -9x² + 72x + 81 in vertex form
B. Correction: y = 2(x-7)(x + 3) = 0.
Divide both sides by 2:
(x - 7)(x + 3) = 0,
x^2 - 4x - 21 = 0.
a. To express the equation y = (x – 3)² - 25 in standard form:
Expand the equation:
y = (x – 3)(x – 3) - 25
y = (x² - 6x + 9) - 25
y = x² - 6x + 9 - 25
y = x² - 6x - 16
Therefore, the equation in standard form is y = x² - 6x - 16.
b. To express the equation y = 2(x – 7)(x + 3) in standard form:
Expand the equation:
y = 2(x² - 4x - 21)
y = 2x² - 8x - 42
Therefore, the equation in standard form is y = 2x² - 8x - 42.
c. To express the equation y = - 2x² + 28x – 26 in factored form:
Factor out the common factor (-2):
y = -2(x² - 14x + 13)
y = -2(x - 1)(x - 13)
Therefore, the equation in factored form is y = -2(x - 1)(x - 13).
d. To express the equation y = -9x² + 72x + 81 in vertex form:
Complete the square by adding and subtracting the square of half the coefficient of x²:
y = -9(x² - 8x) + 81
y = -9(x² - 8x + 16) + 81 - 144
y = -9(x - 4)² - 63
Therefore, the equation in vertex form is y = -9(x - 4)² - 63.
a. To express the equation y = (x – 3)² - 25 in standard form, we need to expand and simplify the equation.
First, let's expand the equation:
y = (x – 3)² - 25
y = (x – 3)(x – 3) - 25
y = (x² – 6x + 9) - 25
y = x² – 6x + 9 - 25
y = x² – 6x – 16
So, the equation in standard form is: y = x² – 6x – 16.
b. To express the equation y = 2(x – 7)(x + 3) in standard form, we need to expand and simplify the equation.
First, let's expand the equation:
y = 2(x – 7)(x + 3)
y = 2(x² – 7x + 3x – 21)
y = 2(x² – 4x – 21)
y = 2x² – 8x – 42
So, the equation in standard form is: y = 2x² – 8x – 42.
c. To express the equation y = -2x² + 28x – 26 in factored form, we need to factor out the common terms.
There is no common factor to factor out in this equation.
So, the equation remains the same: y = -2x² + 28x – 26.
d. To express the equation y = -9x² + 72x + 81 in vertex form, we need to complete the square.
First, let's rearrange the equation:
y = -9x² + 72x + 81
y = -9(x² - 8x) + 81
Now, we need to complete the square inside the parentheses. To do this, we take half of the coefficient of x (-8) and square it.
(-8 / 2)^2 = (-4)^2 = 16
Add 16 inside the parentheses and subtract 16 outside the parentheses to keep the equation balanced:
y = -9(x² - 8x + 16) + 81 - 16
y = -9(x² - 8x + 16) + 65
Now, we can rewrite the equation in vertex form:
y = -9(x - 4)² + 65
So, the equation in vertex form is: y = -9(x - 4)² + 65.
A. Y = (x-3)^2 - 25 = 0
x^2 -6x + 9 - 25 = 0
x^2 - 6x - 16 = 0.
B. y = 2(x - 7)/(x + 3) = 0.
Multiply both sides by x+3:
2(x-7) = 0,
2x - 14 = 0,
x - 7 = 0.
C. y = -2x^2 + 28x - 26 = 0,
Divide both sides by -2:
x^2 - 14x + 13 = 0,
(x - 1)(x - 13) = 0.
2x- 14 = 0.