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

Solve for x (complex solution)

x=\sqrt{877}-11\approx 18.61418579

x=-\left(\sqrt{877}+11\right)\approx -40.61418579

Solve x^2+22x-756=0 | Microsoft Math Solver (1)

Solve for x

x=\sqrt{877}-11\approx 18.61418579

x=-\sqrt{877}-11\approx -40.61418579

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

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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.

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

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QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

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One method is by using inverse operations. This means that you can use addition and subtraction to solve for x if the equation includes multiplication or division. For example, if you have the equation 4x=12, you can divide both sides by 4 to get x=3. Another method for solving for x is by using factoring.

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Why do we solve for x? ›

The algebraic symbol “x” is commonly used to represent any unknown number. If you like, you could say “some number” when you encounter “x” in an equation. You'd only try to determine its value if x represents a number it would be useful to know.

How to solve for x when x is on both sides? ›

Use techniques like the distributive property, combining like terms, factoring, adding or subtracting the same number, and multiplying or dividing by the same non-zero number to isolate "x" and find the answer. Substitute the value of the isolated variable into the other equation.

Where is x =- 1? ›

Algebra Examples

Since x=−1 is a vertical line, there is no y-intercept and the slope is undefined.

What is solve x2 24x =- 80 by completing the square? ›

Summary: By solving x2 - 24x = -80 by completing the square, we get a solution set as {4, 20}.

What is the solution of x2 14x =- 24? ›

By solving by completing the square of the given equation x2 + 14x = -24, we get a solution set as {-12, -2}.

What is the solution to x2 11x 28? ›

We are given that we are required to solve ${x^2} - 11x + 28 = 0$ using the quadratic formula. The general quadratic equation is given by $a{x^2} + bx + c = 0$, where a, b and c are the real numbers. Therefore, the values of x can be 7 and 4. Hence, the roots are 4 and 7.

How to verify answers in maths? ›

Verifying a solution ensures the solution satisfies any equation or inequality by using substitution. Verify whether or not x = 3 is a solution to the conditional equation 2x - 3 = 6 - x. Substitute x = 3 into 2x - 3 = 6 - x to see if a true or false statement results.

How do I know my math answer is correct? ›

How do you check math answers? Back-calculation is the best way for checking your maths exam. The answer that you have got place it in the initial equation and remove some other thing and find it. If the thing that you have found out is same to the one you removed then your answer is correct.

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