Solve x^2+40x-469=0 | Microsoft Math Solver (2024)

x^{2}+40x-469=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{-40±\sqrt{40^{2}-4\left(-469\right)}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 40 for b, and -469 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-40±\sqrt{1600-4\left(-469\right)}}{2}

Square 40.

x=\frac{-40±\sqrt{1600+1876}}{2}

Multiply -4 times -469.

x=\frac{-40±\sqrt{3476}}{2}

Add 1600 to 1876.

x=\frac{-40±2\sqrt{869}}{2}

Take the square root of 3476.

x=\frac{2\sqrt{869}-40}{2}

Now solve the equation x=\frac{-40±2\sqrt{869}}{2} when ± is plus. Add -40 to 2\sqrt{869}.

x=\sqrt{869}-20

Divide -40+2\sqrt{869} by 2.

x=\frac{-2\sqrt{869}-40}{2}

Now solve the equation x=\frac{-40±2\sqrt{869}}{2} when ± is minus. Subtract 2\sqrt{869} from -40.

x=-\sqrt{869}-20

Divide -40-2\sqrt{869} by 2.

x=\sqrt{869}-20 x=-\sqrt{869}-20

The equation is now solved.

x^{2}+40x-469=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}+40x-469-\left(-469\right)=-\left(-469\right)

Add 469 to both sides of the equation.

x^{2}+40x=-\left(-469\right)

Subtracting -469 from itself leaves 0.

x^{2}+40x=469

Subtract -469 from 0.

x^{2}+40x+20^{2}=469+20^{2}

Divide 40, the coefficient of the x term, by 2 to get 20. Then add the square of 20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}+40x+400=469+400

Square 20.

x^{2}+40x+400=869

Add 469 to 400.

\left(x+20\right)^{2}=869

Factor x^{2}+40x+400. 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+20\right)^{2}}=\sqrt{869}

Take the square root of both sides of the equation.

x+20=\sqrt{869} x+20=-\sqrt{869}

Simplify.

x=\sqrt{869}-20 x=-\sqrt{869}-20

Subtract 20 from both sides of the equation.

x ^ 2 +40x -469 = 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 = -40 rs = -469

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 = -20 - u s = -20 + u

Two numbers r and s sum up to -40 exactly when the average of the two numbers is \frac{1}{2}*-40 = -20. 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>

(-20 - u) (-20 + u) = -469

To solve for unknown quantity u, substitute these in the product equation rs = -469

400 - u^2 = -469

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

-u^2 = -469-400 = -869

Simplify the expression by subtracting 400 on both sides

u^2 = 869 u = \pm\sqrt{869} = \pm \sqrt{869}

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

r =-20 - \sqrt{869} = -49.479 s = -20 + \sqrt{869} = 9.479

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}+40x-469=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{-40±\sqrt{40^{2}-4\left(-469\right)}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 40 for b, and -469 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-40±\sqrt{1600-4\left(-469\right)}}{2}

Square 40.

x=\frac{-40±\sqrt{1600+1876}}{2}

Multiply -4 times -469.

x=\frac{-40±\sqrt{3476}}{2}

Add 1600 to 1876.

x=\frac{-40±2\sqrt{869}}{2}

Take the square root of 3476.

x=\frac{2\sqrt{869}-40}{2}

Now solve the equation x=\frac{-40±2\sqrt{869}}{2} when ± is plus. Add -40 to 2\sqrt{869}.

x=\sqrt{869}-20

Divide -40+2\sqrt{869} by 2.

x=\frac{-2\sqrt{869}-40}{2}

Now solve the equation x=\frac{-40±2\sqrt{869}}{2} when ± is minus. Subtract 2\sqrt{869} from -40.

x=-\sqrt{869}-20

Divide -40-2\sqrt{869} by 2.

x=\sqrt{869}-20 x=-\sqrt{869}-20

The equation is now solved.

x^{2}+40x-469=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}+40x-469-\left(-469\right)=-\left(-469\right)

Add 469 to both sides of the equation.

x^{2}+40x=-\left(-469\right)

Subtracting -469 from itself leaves 0.

x^{2}+40x=469

Subtract -469 from 0.

x^{2}+40x+20^{2}=469+20^{2}

Divide 40, the coefficient of the x term, by 2 to get 20. Then add the square of 20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}+40x+400=469+400

Square 20.

x^{2}+40x+400=869

Add 469 to 400.

\left(x+20\right)^{2}=869

Factor x^{2}+40x+400. 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+20\right)^{2}}=\sqrt{869}

Take the square root of both sides of the equation.

x+20=\sqrt{869} x+20=-\sqrt{869}

Simplify.

x=\sqrt{869}-20 x=-\sqrt{869}-20

Subtract 20 from both sides of the equation.