Solve x^2+22x-756=0 | Microsoft Math Solver (2024)

x^{2}+22x-756=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{-22±\sqrt{22^{2}-4\left(-756\right)}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 22 for b, and -756 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-22±\sqrt{484-4\left(-756\right)}}{2}

Square 22.

x=\frac{-22±\sqrt{484+3024}}{2}

Multiply -4 times -756.

x=\frac{-22±\sqrt{3508}}{2}

Add 484 to 3024.

x=\frac{-22±2\sqrt{877}}{2}

Take the square root of 3508.

x=\frac{2\sqrt{877}-22}{2}

Now solve the equation x=\frac{-22±2\sqrt{877}}{2} when ± is plus. Add -22 to 2\sqrt{877}.

x=\sqrt{877}-11

Divide -22+2\sqrt{877} by 2.

x=\frac{-2\sqrt{877}-22}{2}

Now solve the equation x=\frac{-22±2\sqrt{877}}{2} when ± is minus. Subtract 2\sqrt{877} from -22.

x=-\sqrt{877}-11

Divide -22-2\sqrt{877} by 2.

x=\sqrt{877}-11 x=-\sqrt{877}-11

The equation is now solved.

x^{2}+22x-756=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}+22x-756-\left(-756\right)=-\left(-756\right)

Add 756 to both sides of the equation.

x^{2}+22x=-\left(-756\right)

Subtracting -756 from itself leaves 0.

x^{2}+22x=756

Subtract -756 from 0.

x^{2}+22x+11^{2}=756+11^{2}

Divide 22, the coefficient of the x term, by 2 to get 11. Then add the square of 11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}+22x+121=756+121

Square 11.

x^{2}+22x+121=877

Add 756 to 121.

\left(x+11\right)^{2}=877

Factor x^{2}+22x+121. 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+11\right)^{2}}=\sqrt{877}

Take the square root of both sides of the equation.

x+11=\sqrt{877} x+11=-\sqrt{877}

Simplify.

x=\sqrt{877}-11 x=-\sqrt{877}-11

Subtract 11 from both sides of the equation.

x ^ 2 +22x -756 = 0

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

r + s = -22 rs = -756

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

r = -11 - u s = -11 + u

Two numbers r and s sum up to -22 exactly when the average of the two numbers is \frac{1}{2}*-22 = -11. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

(-11 - u) (-11 + u) = -756

To solve for unknown quantity u, substitute these in the product equation rs = -756

121 - u^2 = -756

Simplify by expanding (a -b) (a + b) = a^2 – b^2

-u^2 = -756-121 = -877

Simplify the expression by subtracting 121 on both sides

u^2 = 877 u = \pm\sqrt{877} = \pm \sqrt{877}

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

r =-11 - \sqrt{877} = -40.614 s = -11 + \sqrt{877} = 18.614

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

x^{2}+22x-756=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{-22±\sqrt{22^{2}-4\left(-756\right)}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 22 for b, and -756 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-22±\sqrt{484-4\left(-756\right)}}{2}

Square 22.

x=\frac{-22±\sqrt{484+3024}}{2}

Multiply -4 times -756.

x=\frac{-22±\sqrt{3508}}{2}

Add 484 to 3024.

x=\frac{-22±2\sqrt{877}}{2}

Take the square root of 3508.

x=\frac{2\sqrt{877}-22}{2}

Now solve the equation x=\frac{-22±2\sqrt{877}}{2} when ± is plus. Add -22 to 2\sqrt{877}.

x=\sqrt{877}-11

Divide -22+2\sqrt{877} by 2.

x=\frac{-2\sqrt{877}-22}{2}

Now solve the equation x=\frac{-22±2\sqrt{877}}{2} when ± is minus. Subtract 2\sqrt{877} from -22.

x=-\sqrt{877}-11

Divide -22-2\sqrt{877} by 2.

x=\sqrt{877}-11 x=-\sqrt{877}-11

The equation is now solved.

x^{2}+22x-756=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}+22x-756-\left(-756\right)=-\left(-756\right)

Add 756 to both sides of the equation.

x^{2}+22x=-\left(-756\right)

Subtracting -756 from itself leaves 0.

x^{2}+22x=756

Subtract -756 from 0.

x^{2}+22x+11^{2}=756+11^{2}

Divide 22, the coefficient of the x term, by 2 to get 11. Then add the square of 11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}+22x+121=756+121

Square 11.

x^{2}+22x+121=877

Add 756 to 121.

\left(x+11\right)^{2}=877

Factor x^{2}+22x+121. 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+11\right)^{2}}=\sqrt{877}

Take the square root of both sides of the equation.

x+11=\sqrt{877} x+11=-\sqrt{877}

Simplify.

x=\sqrt{877}-11 x=-\sqrt{877}-11

Subtract 11 from both sides of the equation.