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Math  /  Algebra

Questionctivity 1: Finding Complex Solutions Ive these equations by completing the square. x2−8x+13=0x^{2}-8 x+13=0
2. x2−8x+19=0x^{2}-8 x+19=0

Studdy Solution

STEP 1

1. We are solving quadratic equations by completing the square.
2. The equations are in the form ax2+bx+c=0 ax^2 + bx + c = 0 , where a=1 a = 1 .
3. Completing the square involves creating a perfect square trinomial from the quadratic equation.

STEP 2

1. Solve x2−8x+13=0 x^2 - 8x + 13 = 0 by completing the square.
2. Solve x2−8x+19=0 x^2 - 8x + 19 = 0 by completing the square.

STEP 3

Consider the equation x2−8x+13=0 x^2 - 8x + 13 = 0 . Start by moving the constant term to the other side of the equation:
x2−8x=−13 x^2 - 8x = -13

STEP 4

To complete the square, take half of the coefficient of x x , square it, and add it to both sides. The coefficient of x x is −8-8, so half of it is −4-4, and squaring it gives 1616:
x2−8x+16=−13+16 x^2 - 8x + 16 = -13 + 16

STEP 5

Simplify both sides:
(x−4)2=3 (x - 4)^2 = 3

STEP 6

Take the square root of both sides, remembering to consider both the positive and negative roots:
x−4=±3 x - 4 = \pm \sqrt{3}

STEP 7

Solve for x x by adding 4 to both sides:
x=4±3 x = 4 \pm \sqrt{3}

STEP 8

Consider the equation x2−8x+19=0 x^2 - 8x + 19 = 0 . Start by moving the constant term to the other side of the equation:
x2−8x=−19 x^2 - 8x = -19

STEP 9

To complete the square, take half of the coefficient of x x , square it, and add it to both sides. The coefficient of x x is −8-8, so half of it is −4-4, and squaring it gives 1616:
x2−8x+16=−19+16 x^2 - 8x + 16 = -19 + 16

STEP 10

Simplify both sides:
(x−4)2=−3 (x - 4)^2 = -3

STEP 11

Take the square root of both sides, remembering to consider both the positive and negative roots. Since we have a negative number under the square root, we will use imaginary numbers:
x−4=±−3 x - 4 = \pm \sqrt{-3} x−4=±i3 x - 4 = \pm i\sqrt{3}

STEP 12

Solve for x x by adding 4 to both sides:
x=4±i3 x = 4 \pm i\sqrt{3}
The solutions for the equations are:
1. x=4±3 x = 4 \pm \sqrt{3}
2. x=4±i3 x = 4 \pm i\sqrt{3}

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