Solve x^2-1060x+19072464=0 | Microsoft Math Solver (2024)

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x^{2}-1060x+19072464=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(-1060\right)±\sqrt{\left(-1060\right)^{2}-4\times 19072464}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1060 for b, and 19072464 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-\left(-1060\right)±\sqrt{1123600-4\times 19072464}}{2}

Square -1060.

x=\frac{-\left(-1060\right)±\sqrt{1123600-76289856}}{2}

Multiply -4 times 19072464.

x=\frac{-\left(-1060\right)±\sqrt{-75166256}}{2}

Add 1123600 to -76289856.

x=\frac{-\left(-1060\right)±4\sqrt{4697891}i}{2}

Take the square root of -75166256.

x=\frac{1060±4\sqrt{4697891}i}{2}

The opposite of -1060 is 1060.

x=\frac{1060+4\sqrt{4697891}i}{2}

Now solve the equation x=\frac{1060±4\sqrt{4697891}i}{2} when ± is plus. Add 1060 to 4i\sqrt{4697891}.

x=530+2\sqrt{4697891}i

Divide 1060+4i\sqrt{4697891} by 2.

x=\frac{-4\sqrt{4697891}i+1060}{2}

Now solve the equation x=\frac{1060±4\sqrt{4697891}i}{2} when ± is minus. Subtract 4i\sqrt{4697891} from 1060.

x=-2\sqrt{4697891}i+530

Divide 1060-4i\sqrt{4697891} by 2.

x=530+2\sqrt{4697891}i x=-2\sqrt{4697891}i+530

The equation is now solved.

x^{2}-1060x+19072464=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}-1060x+19072464-19072464=-19072464

Subtract 19072464 from both sides of the equation.

x^{2}-1060x=-19072464

Subtracting 19072464 from itself leaves 0.

x^{2}-1060x+\left(-530\right)^{2}=-19072464+\left(-530\right)^{2}

Divide -1060, the coefficient of the x term, by 2 to get -530. Then add the square of -530 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}-1060x+280900=-19072464+280900

Square -530.

x^{2}-1060x+280900=-18791564

Add -19072464 to 280900.

\left(x-530\right)^{2}=-18791564

Factor x^{2}-1060x+280900. 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-530\right)^{2}}=\sqrt{-18791564}

Take the square root of both sides of the equation.

x-530=2\sqrt{4697891}i x-530=-2\sqrt{4697891}i

Simplify.

x=530+2\sqrt{4697891}i x=-2\sqrt{4697891}i+530

Add 530 to both sides of the equation.

x ^ 2 -1060x +19072464 = 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 = 1060 rs = 19072464

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 = 530 - u s = 530 + u

Two numbers r and s sum up to 1060 exactly when the average of the two numbers is \frac{1}{2}*1060 = 530. 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>

(530 - u) (530 + u) = 19072464

To solve for unknown quantity u, substitute these in the product equation rs = 19072464

280900 - u^2 = 19072464

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

-u^2 = 19072464-280900 = 18791564

Simplify the expression by subtracting 280900 on both sides

u^2 = -18791564 u = \pm\sqrt{-18791564} = \pm \sqrt{18791564}i

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

r =530 - \sqrt{18791564}i s = 530 + \sqrt{18791564}i

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