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

Solve for x

x=\frac{2\sqrt{329}}{3}+14\approx 26.092238098

x=-\frac{2\sqrt{329}}{3}+14\approx 1.907761902

Solve 27x^2-756x+1344=0 | Microsoft Math Solver (1)



Quadratic Equation5 problems similar to: 27 x ^ { 2 } - 756 x + 1344 = 0

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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.


Multiply 2 times 27.


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


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


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


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.


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.


Subtract 1344 from both sides of the equation.


Subtracting 1344 from itself leaves 0.


Divide both sides by 27.


Dividing by 27 undoes the multiplication by 27.


Divide -756 by 27.


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


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.


Square -14.


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


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}.


Take the square root of both sides of the equation.

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


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='' 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} = 1.908 s = 14 + \frac{\sqrt{1316}}{3} = 26.092

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

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