Solve for x

x=4

x=7

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a+b=-11 ab=28

To solve the equation, factor x^{2}-11x+28 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

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

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 28.

-1-28=-29 -2-14=-16 -4-7=-11

Calculate the sum for each pair.

a=-7 b=-4

The solution is the pair that gives sum -11.

\left(x-7\right)\left(x-4\right)

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

x=7 x=4

To find equation solutions, solve x-7=0 and x-4=0.

a+b=-11 ab=1\times 28=28

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

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

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 28.

-1-28=-29 -2-14=-16 -4-7=-11

Calculate the sum for each pair.

a=-7 b=-4

The solution is the pair that gives sum -11.

\left(x^{2}-7x\right)+\left(-4x+28\right)

Rewrite x^{2}-11x+28 as \left(x^{2}-7x\right)+\left(-4x+28\right).

x\left(x-7\right)-4\left(x-7\right)

Factor out x in the first and -4 in the second group.

\left(x-7\right)\left(x-4\right)

Factor out common term x-7 by using distributive property.

x=7 x=4

To find equation solutions, solve x-7=0 and x-4=0.

x^{2}-11x+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{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 28}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-\left(-11\right)±\sqrt{121-4\times 28}}{2}

Square -11.

x=\frac{-\left(-11\right)±\sqrt{121-112}}{2}

Multiply -4 times 28.

x=\frac{-\left(-11\right)±\sqrt{9}}{2}

Add 121 to -112.

x=\frac{-\left(-11\right)±3}{2}

Take the square root of 9.

x=\frac{11±3}{2}

The opposite of -11 is 11.

x=\frac{14}{2}

Now solve the equation x=\frac{11±3}{2} when ± is plus. Add 11 to 3.

x=7

Divide 14 by 2.

x=\frac{8}{2}

Now solve the equation x=\frac{11±3}{2} when ± is minus. Subtract 3 from 11.

x=4

Divide 8 by 2.

x=7 x=4

The equation is now solved.

x^{2}-11x+28=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.

x^{2}-11x+28-28=-28

Subtract 28 from both sides of the equation.

x^{2}-11x=-28

Subtracting 28 from itself leaves 0.

x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-28+\left(-\frac{11}{2}\right)^{2}

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}-11x+\frac{121}{4}=-28+\frac{121}{4}

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

x^{2}-11x+\frac{121}{4}=\frac{9}{4}

Add -28 to \frac{121}{4}.

\left(x-\frac{11}{2}\right)^{2}=\frac{9}{4}

Factor x^{2}-11x+\frac{121}{4}. 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-\frac{11}{2}\right)^{2}}=\sqrt{\frac{9}{4}}

Take the square root of both sides of the equation.

x-\frac{11}{2}=\frac{3}{2} x-\frac{11}{2}=-\frac{3}{2}

Simplify.

x=7 x=4

Add \frac{11}{2} to both sides of the equation.

x ^ 2 -11x +28 = 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.

r + s = 11 rs = 28

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 = \frac{11}{2} - u s = \frac{11}{2} + u

Two numbers r and s sum up to 11 exactly when the average of the two numbers is \frac{1}{2}*11 = \frac{11}{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>

(\frac{11}{2} - u) (\frac{11}{2} + u) = 28

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

\frac{121}{4} - u^2 = 28

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

-u^2 = 28-\frac{121}{4} = -\frac{9}{4}

Simplify the expression by subtracting \frac{121}{4} on both sides

u^2 = \frac{9}{4} u = \pm\sqrt{\frac{9}{4}} = \pm \frac{3}{2}

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

r =\frac{11}{2} - \frac{3}{2} = 4 s = \frac{11}{2} + \frac{3}{2} = 7

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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