Solve x^2+27x-28=

$$x ^ { 2 } + 27 x - 28 =$$

Factor

$$\left(x-1\right)\left(x+28\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-28. To find a and b, set up a system to be solved.

$$a+b=27 ab=1\left(-28\right)=-28$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -28.

$$-1,28 -2,14 -4,7$$

Calculate the sum for each pair.

$$-1+28=27 -2+14=12 -4+7=3$$

The solution is the pair that gives sum 27.

$$a=-1 b=28$$

Rewrite x^{2}+27x-28 as \left(x^{2}-x\right)+\left(28x-28\right).

$$\left(x^{2}-x\right)+\left(28x-28\right)$$

Factor out x in the first and 28 in the second group.

$$x\left(x-1\right)+28\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+28\right)$$

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.

$$x^{2}+27x-28=0$$

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.

$$x=\frac{-27±\sqrt{27^{2}-4\left(-28\right)}}{2}$$

Square 27.

$$x=\frac{-27±\sqrt{729-4\left(-28\right)}}{2}$$

Multiply -4 times -28.

$$x=\frac{-27±\sqrt{729+112}}{2}$$

Add 729 to 112.

$$x=\frac{-27±\sqrt{841}}{2}$$

Take the square root of 841.

$$x=\frac{-27±29}{2}$$

Now solve the equation x=\frac{-27±29}{2} when ± is plus. Add -27 to 29.

$$x=\frac{2}{2}$$

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-27±29}{2} when ± is minus. Subtract 29 from -27.

$$x=-\frac{56}{2}$$

Divide -56 by 2.

$$x=-28$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -28 for x_{2}.

$$x^{2}+27x-28=\left(x-1\right)\left(x-\left(-28\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}+27x-28=\left(x-1\right)\left(x+28\right)$$

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.

$$x ^ 2 +27x -28 = 0$$

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

$$r + s = -27 rs = -28$$

Two numbers r and s sum up to -27 exactly when the average of the two numbers is \frac{1}{2}*-27 = -\frac{27}{2}. 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>

$$r = -\frac{27}{2} - u s = -\frac{27}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -28

$$(-\frac{27}{2} - u) (-\frac{27}{2} + u) = -28$$

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

$$\frac{729}{4} - u^2 = -28$$

Simplify the expression by subtracting \frac{729}{4} on both sides

$$-u^2 = -28-\frac{729}{4} = -\frac{841}{4}$$

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

$$u^2 = \frac{841}{4} u = \pm\sqrt{\frac{841}{4}} = \pm \frac{29}{2}$$

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

$$r =-\frac{27}{2} - \frac{29}{2} = -28 s = -\frac{27}{2} + \frac{29}{2} = 1$$

Evaluate

$$\left(x-1\right)\left(x+28\right)$$