Solve 27x^2-756x+1344=0 | Microsoft Math Solver (2024)

27x^{2}-756x+1344=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(-756\right)±\sqrt{\left(-756\right)^{2}-4\times 27\times 1344}}{2\times 27}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 27 for a, -756 for b, and 1344 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-\left(-756\right)±\sqrt{571536-4\times 27\times 1344}}{2\times 27}

Square -756.

x=\frac{-\left(-756\right)±\sqrt{571536-108\times 1344}}{2\times 27}

Multiply -4 times 27.

x=\frac{-\left(-756\right)±\sqrt{571536-145152}}{2\times 27}

Multiply -108 times 1344.

x=\frac{-\left(-756\right)±\sqrt{426384}}{2\times 27}

Add 571536 to -145152.

x=\frac{-\left(-756\right)±36\sqrt{329}}{2\times 27}

Take the square root of 426384.

x=\frac{756±36\sqrt{329}}{2\times 27}

The opposite of -756 is 756.

x=\frac{756±36\sqrt{329}}{54}

Multiply 2 times 27.

x=\frac{36\sqrt{329}+756}{54}

Now solve the equation x=\frac{756±36\sqrt{329}}{54} when ± is plus. Add 756 to 36\sqrt{329}.

x=\frac{2\sqrt{329}}{3}+14

Divide 756+36\sqrt{329} by 54.

x=\frac{756-36\sqrt{329}}{54}

Now solve the equation x=\frac{756±36\sqrt{329}}{54} when ± is minus. Subtract 36\sqrt{329} from 756.

x=-\frac{2\sqrt{329}}{3}+14

Divide 756-36\sqrt{329} by 54.

x=\frac{2\sqrt{329}}{3}+14 x=-\frac{2\sqrt{329}}{3}+14

The equation is now solved.

27x^{2}-756x+1344=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.

27x^{2}-756x+1344-1344=-1344

Subtract 1344 from both sides of the equation.

27x^{2}-756x=-1344

Subtracting 1344 from itself leaves 0.

\frac{27x^{2}-756x}{27}=-\frac{1344}{27}

Divide both sides by 27.

x^{2}+\left(-\frac{756}{27}\right)x=-\frac{1344}{27}

Dividing by 27 undoes the multiplication by 27.

x^{2}-28x=-\frac{1344}{27}

Divide -756 by 27.

x^{2}-28x=-\frac{448}{9}

Reduce the fraction \frac{-1344}{27} to lowest terms by extracting and canceling out 3.

x^{2}-28x+\left(-14\right)^{2}=-\frac{448}{9}+\left(-14\right)^{2}

Divide -28, the coefficient of the x term, by 2 to get -14. Then add the square of -14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}-28x+196=-\frac{448}{9}+196

Square -14.

x^{2}-28x+196=\frac{1316}{9}

Add -\frac{448}{9} to 196.

\left(x-14\right)^{2}=\frac{1316}{9}

Factor x^{2}-28x+196. 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-14\right)^{2}}=\sqrt{\frac{1316}{9}}

Take the square root of both sides of the equation.

x-14=\frac{2\sqrt{329}}{3} x-14=-\frac{2\sqrt{329}}{3}

Simplify.

x=\frac{2\sqrt{329}}{3}+14 x=-\frac{2\sqrt{329}}{3}+14

Add 14 to both sides of the equation.

x ^ 2 -28x +\frac{448}{9} = 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.This is achieved by dividing both sides of the equation by 27

r + s = 28 rs = \frac{448}{9}

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 = 14 - u s = 14 + u

Two numbers r and s sum up to 28 exactly when the average of the two numbers is \frac{1}{2}*28 = 14. 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>

(14 - u) (14 + u) = \frac{448}{9}

To solve for unknown quantity u, substitute these in the product equation rs = \frac{448}{9}

196 - u^2 = \frac{448}{9}

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

-u^2 = \frac{448}{9}-196 = -\frac{1316}{9}

Simplify the expression by subtracting 196 on both sides

u^2 = \frac{1316}{9} u = \pm\sqrt{\frac{1316}{9}} = \pm \frac{\sqrt{1316}}{3}

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

r =14 - \frac{\sqrt{1316}}{3} = 1.907761901855297 s = 14 + \frac{\sqrt{1316}}{3} = 26.092238098144705

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