12x2-96x+144=0Two solutions were found : x = 6 x = 2 Step by step solution : Step 1 :Equation at the end of step 1 : ((22•3x2) - 96x) + 144 = 0 Step 2 : Step 3 :Pulling out like terms ...
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.
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}.
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.
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}.
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} = 1.908 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.
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