Solve the equation. Use factoring or the quadratic formula, whichever is appropriate. Try factoring first. If you have any difficulty factoring, then go right to the quadratic formula. (Enter your answers as a comma-separated list.)
(x+2)^2 + (x-7)(x-2)=16
(x+2)^2 + (x-7)(x-2) = 16
x^2 + 4x + 4 + x^2 - 2x -7x + 14 = 16
2x^2 - 5x + 18 = 16
2x^2 - 5x + 2 = 0
A*C = 2*2 = 4 = (-1)*(-4).
Choose the pair of factors whose sum = B
2x^2 - (x+4x) +2 = 0
(2x^2-x) - (4x-2) = 0
x(2x-1) - 2(2x-1) = 0
(2x-1)(x-2) = 0
2x-1 = 0
2x = 1
X = 1/2
x-2 = 0
X = 2
To solve the equation (x+2)^2 + (x-7)(x-2) = 16, let's expand and simplify:
(x+2)^2 + (x-7)(x-2) = 16
(x+2)(x+2) + (x-7)(x-2) = 16
Using the distributive property, we can multiply the two binomials:
(x^2 + 2x + 2x + 4) + (x^2 - 2x - 7x + 14) = 16
Simplifying further:
x^2 + 4x + 4 + x^2 - 9x + 14 = 16
Combining like terms:
2x^2 - 5x + 18 = 16
Rearranging to form a quadratic equation in standard form:
2x^2 - 5x + 18 - 16 = 0
2x^2 - 5x + 2 = 0
Now, we can proceed to solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = 2, b = -5, and c = 2.
x = (-(-5) ± √((-5)^2 - 4(2)(2))) / (2(2))
x = (5 ± √(25 - 16)) / 4
x = (5 ± √9) / 4
x = (5 ± 3) / 4
Now, we can find the two possible values of x:
x₁ = (5 + 3) / 4 = 8 / 4 = 2
x₂ = (5 - 3) / 4 = 2 / 4 = 1/2
Therefore, the solutions to the equation (x+2)^2 + (x-7)(x-2) = 16 are x = 2 and x = 1/2.
To solve the equation (x+2)^2 + (x-7)(x-2) = 16, we can first simplify the equation and then use factoring or the quadratic formula to find the values of x.
Let's simplify the equation:
(x+2)^2 + (x-7)(x-2) = 16
Expanding the square (x+2)^2, we get:
(x+2)(x+2) + (x-7)(x-2) = 16
(x+2)(x+2) + (x-7)(x-2) can be further expanded as:
(x+2)(x+2) + (x^2 - 2x - 7x + 14) = 16
(x+2)(x+2) + (x^2 - 9x + 14) = 16
Now we can simplify further:
(x+2)^2 + x^2 - 9x + 14 = 16
(x^2 + 4x + 4) + x^2 - 9x + 14 = 16
x^2 + 4x + 4 + x^2 - 9x + 14 = 16
Combining like terms, we obtain:
2x^2 - 5x + 18 = 16
Now we can proceed to solve the equation using factoring or the quadratic formula. Let's try factoring first:
To factor the quadratic equation 2x^2 - 5x + 18 = 16, we need to find two binomials that multiply together to give the quadratic equation.
Since the equation is not easily factorable, we can go ahead and use the quadratic formula.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 2, b = -5, and c = 18 - 16 = 2.
Substituting these values into the quadratic formula, we have:
x = (-(-5) ± √((-5)^2 - 4 * 2 * 2)) / (2 * 2)
Simplifying:
x = (5 ± √(25 - 16)) / 4
x = (5 ± √9) / 4
x = (5 ± 3) / 4
So, we have two possible solutions:
x = (5 + 3) / 4 = 8 / 4 = 2
x = (5 - 3) / 4 = 2 / 4 = 1/2
Therefore, the solutions to the equation (x+2)^2 + (x-7)(x-2) = 16 are x = 2 and x = 1/2.