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

a+b=-1 ab=-756

To solve the equation, factor x^{2}-x-756 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,-756 2,-378 3,-252 4,-189 6,-126 7,-108 9,-84 12,-63 14,-54 18,-42 21,-36 27,-28

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -756.

1-756=-755 2-378=-376 3-252=-249 4-189=-185 6-126=-120 7-108=-101 9-84=-75 12-63=-51 14-54=-40 18-42=-24 21-36=-15 27-28=-1

Calculate the sum for each pair.

a=-28 b=27

The solution is the pair that gives sum -1.

\left(x-28\right)\left(x+27\right)

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

x=28 x=-27

To find equation solutions, solve x-28=0 and x+27=0.

a+b=-1 ab=1\left(-756\right)=-756

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-756. To find a and b, set up a system to be solved.

1,-756 2,-378 3,-252 4,-189 6,-126 7,-108 9,-84 12,-63 14,-54 18,-42 21,-36 27,-28

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -756.

1-756=-755 2-378=-376 3-252=-249 4-189=-185 6-126=-120 7-108=-101 9-84=-75 12-63=-51 14-54=-40 18-42=-24 21-36=-15 27-28=-1

Calculate the sum for each pair.

a=-28 b=27

The solution is the pair that gives sum -1.

x\left(x-28\right)+27\left(x-28\right)

Factor out x in the first and 27 in the second group.

\left(x-28\right)\left(x+27\right)

Factor out common term x-28 by using distributive property.

x=28 x=-27

To find equation solutions, solve x-28=0 and x+27=0.

x^{2}-x-756=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(-1\right)±\sqrt{1-4\left(-756\right)}}{2}

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -756 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

x=\frac{-\left(-1\right)±\sqrt{1+3024}}{2}

Multiply -4 times -756.

x=\frac{-\left(-1\right)±\sqrt{3025}}{2}

Add 1 to 3024.

x=\frac{-\left(-1\right)±55}{2}

Take the square root of 3025.

x=\frac{1±55}{2}

The opposite of -1 is 1.

x=\frac{56}{2}

Now solve the equation x=\frac{1±55}{2} when ± is plus. Add 1 to 55.

x=-\frac{54}{2}

Now solve the equation x=\frac{1±55}{2} when ± is minus. Subtract 55 from 1.

x=-27

Divide -54 by 2.

x=28 x=-27

The equation is now solved.

x^{2}-x-756=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}-x-756-\left(-756\right)=-\left(-756\right)

Add 756 to both sides of the equation.

x^{2}-x=-\left(-756\right)

Subtracting -756 from itself leaves 0.

x^{2}-x=756

Subtract -756 from 0.

x^{2}-x+\left(-\frac{1}{2}\right)^{2}=756+\left(-\frac{1}{2}\right)^{2}

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

x^{2}-x+\frac{1}{4}=756+\frac{1}{4}

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

x^{2}-x+\frac{1}{4}=\frac{3025}{4}

Add 756 to \frac{1}{4}.

\left(x-\frac{1}{2}\right)^{2}=\frac{3025}{4}

Factor x^{2}-x+\frac{1}{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{1}{2}\right)^{2}}=\sqrt{\frac{3025}{4}}

Take the square root of both sides of the equation.

x-\frac{1}{2}=\frac{55}{2} x-\frac{1}{2}=-\frac{55}{2}

Simplify.

x=28 x=-27

Add \frac{1}{2} to both sides of the equation.

x ^ 2 -1x -756 = 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 = 1 rs = -756

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{1}{2} - u s = \frac{1}{2} + u

Two numbers r and s sum up to 1 exactly when the average of the two numbers is \frac{1}{2}*1 = \frac{1}{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{1}{2} - u) (\frac{1}{2} + u) = -756

To solve for unknown quantity u, substitute these in the product equation rs = -756

\frac{1}{4} - u^2 = -756

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

-u^2 = -756-\frac{1}{4} = -\frac{3025}{4}

Simplify the expression by subtracting \frac{1}{4} on both sides

u^2 = \frac{3025}{4} u = \pm\sqrt{\frac{3025}{4}} = \pm \frac{55}{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{1}{2} - \frac{55}{2} = -27 s = \frac{1}{2} + \frac{55}{2} = 28

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

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