Solve for x
x=\sqrt{51}+6\approx 13.141428429
x=6-\sqrt{51}\approx -1.141428429
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x^{2}-12x-15=0
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.
x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-15\right)}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-12\right)±\sqrt{144-4\left(-15\right)}}{2}
Square -12.
x=\frac{-\left(-12\right)±\sqrt{144+60}}{2}
Multiply -4 times -15.
x=\frac{-\left(-12\right)±\sqrt{204}}{2}
Add 144 to 60.
x=\frac{-\left(-12\right)±2\sqrt{51}}{2}
Take the square root of 204.
x=\frac{12±2\sqrt{51}}{2}
The opposite of -12 is 12.
x=\frac{2\sqrt{51}+12}{2}
Now solve the equation x=\frac{12±2\sqrt{51}}{2} when ± is plus. Add 12 to 2\sqrt{51}.
x=\sqrt{51}+6
Divide 12+2\sqrt{51} by 2.
x=\frac{12-2\sqrt{51}}{2}
Now solve the equation x=\frac{12±2\sqrt{51}}{2} when ± is minus. Subtract 2\sqrt{51} from 12.
x=6-\sqrt{51}
Divide 12-2\sqrt{51} by 2.
x=\sqrt{51}+6 x=6-\sqrt{51}
The equation is now solved.
x^{2}-12x-15=0
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.
x^{2}-12x-15-\left(-15\right)=-\left(-15\right)
Add 15 to both sides of the equation.
x^{2}-12x=-\left(-15\right)
Subtracting -15 from itself leaves 0.
x^{2}-12x=15
Subtract -15 from 0.
x^{2}-12x+\left(-6\right)^{2}=15+\left(-6\right)^{2}
Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-12x+36=15+36
Square -6.
x^{2}-12x+36=51
Add 15 to 36.
\left(x-6\right)^{2}=51
Factor x^{2}-12x+36. 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(x-6\right)^{2}}=\sqrt{51}
Take the square root of both sides of the equation.
x-6=\sqrt{51} x-6=-\sqrt{51}
Simplify.
x=\sqrt{51}+6 x=6-\sqrt{51}
Add 6 to both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}