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Solve for a (complex solution)
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a^{2}+8a=13
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
a^{2}+8a-13=13-13
Subtract 13 from both sides of the equation.
a^{2}+8a-13=0
Subtracting 13 from itself leaves 0.
a=\frac{-8±\sqrt{8^{2}-4\left(-13\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-8±\sqrt{64-4\left(-13\right)}}{2}
Square 8.
a=\frac{-8±\sqrt{64+52}}{2}
Multiply -4 times -13.
a=\frac{-8±\sqrt{116}}{2}
Add 64 to 52.
a=\frac{-8±2\sqrt{29}}{2}
Take the square root of 116.
a=\frac{2\sqrt{29}-8}{2}
Now solve the equation a=\frac{-8±2\sqrt{29}}{2} when ± is plus. Add -8 to 2\sqrt{29}.
a=\sqrt{29}-4
Divide -8+2\sqrt{29} by 2.
a=\frac{-2\sqrt{29}-8}{2}
Now solve the equation a=\frac{-8±2\sqrt{29}}{2} when ± is minus. Subtract 2\sqrt{29} from -8.
a=-\sqrt{29}-4
Divide -8-2\sqrt{29} by 2.
a=\sqrt{29}-4 a=-\sqrt{29}-4
The equation is now solved.
a^{2}+8a=13
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
a^{2}+8a+4^{2}=13+4^{2}
Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
a^{2}+8a+16=13+16
Square 4.
a^{2}+8a+16=29
Add 13 to 16.
\left(a+4\right)^{2}=29
Factor a^{2}+8a+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(a+4\right)^{2}}=\sqrt{29}
Take the square root of both sides of the equation.
a+4=\sqrt{29} a+4=-\sqrt{29}
Simplify.
a=\sqrt{29}-4 a=-\sqrt{29}-4
Subtract 4 from both sides of the equation.
a^{2}+8a=13
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
a^{2}+8a-13=13-13
Subtract 13 from both sides of the equation.
a^{2}+8a-13=0
Subtracting 13 from itself leaves 0.
a=\frac{-8±\sqrt{8^{2}-4\left(-13\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
a=\frac{-8±\sqrt{64-4\left(-13\right)}}{2}
Square 8.
a=\frac{-8±\sqrt{64+52}}{2}
Multiply -4 times -13.
a=\frac{-8±\sqrt{116}}{2}
Add 64 to 52.
a=\frac{-8±2\sqrt{29}}{2}
Take the square root of 116.
a=\frac{2\sqrt{29}-8}{2}
Now solve the equation a=\frac{-8±2\sqrt{29}}{2} when ± is plus. Add -8 to 2\sqrt{29}.
a=\sqrt{29}-4
Divide -8+2\sqrt{29} by 2.
a=\frac{-2\sqrt{29}-8}{2}
Now solve the equation a=\frac{-8±2\sqrt{29}}{2} when ± is minus. Subtract 2\sqrt{29} from -8.
a=-\sqrt{29}-4
Divide -8-2\sqrt{29} by 2.
a=\sqrt{29}-4 a=-\sqrt{29}-4
The equation is now solved.
a^{2}+8a=13
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
a^{2}+8a+4^{2}=13+4^{2}
Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
a^{2}+8a+16=13+16
Square 4.
a^{2}+8a+16=29
Add 13 to 16.
\left(a+4\right)^{2}=29
Factor a^{2}+8a+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(a+4\right)^{2}}=\sqrt{29}
Take the square root of both sides of the equation.
a+4=\sqrt{29} a+4=-\sqrt{29}
Simplify.
a=\sqrt{29}-4 a=-\sqrt{29}-4
Subtract 4 from both sides of the equation.