Solve -4x^2+133x-63

$$- 4 x ^ { 2 } + 133 x - 63$$

Factor

$$-4\left(x-\frac{133-\sqrt{16681}}{8}\right)\left(x-\frac{\sqrt{16681}+133}{8}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-4x^{2}+133x-63=0$$

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.

$$x=\frac{-133±\sqrt{133^{2}-4\left(-4\right)\left(-63\right)}}{2\left(-4\right)}$$

Square 133.

$$x=\frac{-133±\sqrt{17689-4\left(-4\right)\left(-63\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-133±\sqrt{17689+16\left(-63\right)}}{2\left(-4\right)}$$

Multiply 16 times -63.

$$x=\frac{-133±\sqrt{17689-1008}}{2\left(-4\right)}$$

Add 17689 to -1008.

$$x=\frac{-133±\sqrt{16681}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-133±\sqrt{16681}}{-8}$$

Now solve the equation x=\frac{-133±\sqrt{16681}}{-8} when ± is plus. Add -133 to \sqrt{16681}.

$$x=\frac{\sqrt{16681}-133}{-8}$$

Divide -133+\sqrt{16681} by -8.

$$x=\frac{133-\sqrt{16681}}{8}$$

Now solve the equation x=\frac{-133±\sqrt{16681}}{-8} when ± is minus. Subtract \sqrt{16681} from -133.

$$x=\frac{-\sqrt{16681}-133}{-8}$$

Divide -133-\sqrt{16681} by -8.

$$x=\frac{\sqrt{16681}+133}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{133-\sqrt{16681}}{8} for x_{1} and \frac{133+\sqrt{16681}}{8} for x_{2}.

$$-4x^{2}+133x-63=-4\left(x-\frac{133-\sqrt{16681}}{8}\right)\left(x-\frac{\sqrt{16681}+133}{8}\right)$$

Steps Using Direct Factoring Method

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.

$$x ^ 2 -\frac{133}{4}x +\frac{63}{4} = 0$$

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

$$r + s = \frac{133}{4} rs = \frac{63}{4}$$

Two numbers r and s sum up to \frac{133}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{133}{4} = \frac{133}{8}. 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>

$$r = \frac{133}{8} - u s = \frac{133}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{63}{4}

$$(\frac{133}{8} - u) (\frac{133}{8} + u) = \frac{63}{4}$$

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

$$\frac{17689}{64} - u^2 = \frac{63}{4}$$

Simplify the expression by subtracting \frac{17689}{64} on both sides

$$-u^2 = \frac{63}{4}-\frac{17689}{64} = -\frac{16681}{64}$$

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

$$u^2 = \frac{16681}{64} u = \pm\sqrt{\frac{16681}{64}} = \pm \frac{\sqrt{16681}}{8}$$

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

$$r =\frac{133}{8} - \frac{\sqrt{16681}}{8} = 0.481 s = \frac{133}{8} + \frac{\sqrt{16681}}{8} = 32.769$$

Evaluate

$$-4x^{2}+133x-63$$