Solve the quadratic equation by completing the square.
x^2 + 14x + 35 = 0
and
x^2 + 8x = 5
I will do the first, you show me how you did the second, ok?
x^2 + 14x + .... = -35 + .....
x^2 + 14x + 49 = -35 + 49
(x+7)^2 = 14
x+7 = ± √14
x = -7 ± √14
To solve the quadratic equation by completing the square, follow these steps:
1. Quadratic equation: x^2 + 14x + 35 = 0
2. Divide the entire equation by the coefficient of x^2 to make the coefficient 1:
(x^2 + 14x + 35)/1 = 0
3. To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation:
x^2 + 14x + (14/2)^2 = -35 + (14/2)^2
Simplifying, we have:
x^2 + 14x + 49 = -35 + 49
4. Simplify the right side of the equation:
x^2 + 14x + 49 = 14
5. Write the left side of the equation as a perfect square:
(x + 7)^2 = 14
6. Take the square root of both sides of the equation:
√(x + 7)^2 = √14
7. Solve for x:
x + 7 = ±√14
8. Subtract 7 from both sides:
x = -7 ±√14
Therefore, the solutions to the quadratic equation x^2 + 14x + 35 = 0 are x = -7 + √14 and x = -7 - √14.
Now let's solve the second equation:
1. Quadratic equation: x^2 + 8x = 5
2. Move the constant term (5) to the right side of the equation:
x^2 + 8x - 5 = 0
3. Divide the entire equation by the coefficient of x^2 to make the coefficient 1:
(x^2 + 8x - 5)/1 = 0
4. To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation:
x^2 + 8x + (8/2)^2 = 5 + (8/2)^2
Simplifying, we have:
x^2 + 8x + 16 = 5 + 16
5. Simplify the right side of the equation:
x^2 + 8x + 16 = 21
6. Write the left side of the equation as a perfect square:
(x + 4)^2 = 21
7. Take the square root of both sides of the equation:
√(x + 4)^2 = √21
8. Solve for x:
x + 4 = ±√21
9. Subtract 4 from both sides:
x = -4 ± √21
Therefore, the solutions to the quadratic equation x^2 + 8x = 5 are x = -4 + √21 and x = -4 - √21.
To solve a quadratic equation by completing the square, follow these steps:
1. Write the equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.
For the equation x^2 + 14x + 35 = 0:
a = 1, b = 14, c = 35
For the equation x^2 + 8x = 5:
a = 1, b = 8, c = -5 (move the constant term to the other side of the equation)
2. Rewrite the equation with the constant term on the right side:
For x^2 + 14x + 35 = 0:
x^2 + 14x = -35
For x^2 + 8x - 5 = 0:
x^2 + 8x = 5
3. Add the square of half the coefficient of x to both sides of the equation:
For x^2 + 14x = -35:
x^2 + 14x + (14/2)^2 = -35 + (14/2)^2
x^2 + 14x + 49 = -35 + 49
x^2 + 14x + 49 = 14
For x^2 + 8x = 5:
x^2 + 8x + (8/2)^2 = 5 + (8/2)^2
x^2 + 8x + 16 = 5 + 16
x^2 + 8x + 16 = 21
4. Factor the perfect square trinomial on the left side:
For x^2 + 14x + 49 = 14:
(x + 7)^2 = 14
For x^2 + 8x + 16 = 21:
(x + 4)^2 = 21
5. Take the square root of both sides of the equation:
For (x + 7)^2 = 14:
x + 7 = ±√14
For (x + 4)^2 = 21:
x + 4 = ±√21
6. Solve for x by subtracting the constant term on the left side:
For x + 7 = ±√14:
x = -7 ± √14
For x + 4 = ±√21:
x = -4 ± √21
So, the solutions for the equation x^2 + 14x + 35 = 0 are x = -7 ± √14, and the solutions for the equation x^2 + 8x = 5 are x = -4 ± √21.