Solve varphi^2-10varphi+21=0

$$\varphi ^ { 2 } - 10 \varphi + 21 = 0$$

Solve for φ

$$\phi =3$$

$$\phi =7$$

Steps Using Factoring

To solve the equation, factor \phi ^{2}-10\phi +21 using formula \phi ^{2}+\left(a+b\right)\phi +ab=\left(\phi +a\right)\left(\phi +b\right). To find a and b, set up a system to be solved.

$$a+b=-10 ab=21$$

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 21.

$$-1,-21 -3,-7$$

Calculate the sum for each pair.

$$-1-21=-22 -3-7=-10$$

The solution is the pair that gives sum -10.

$$a=-7 b=-3$$

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

$$\left(\phi -7\right)\left(\phi -3\right)$$

To find equation solutions, solve \phi -7=0 and \phi -3=0.

$$\phi =7 \phi =3$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as \phi ^{2}+a\phi +b\phi +21. To find a and b, set up a system to be solved.

$$a+b=-10 ab=1\times 21=21$$

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 21.

$$-1,-21 -3,-7$$

Calculate the sum for each pair.

$$-1-21=-22 -3-7=-10$$

The solution is the pair that gives sum -10.

$$a=-7 b=-3$$

Rewrite \phi ^{2}-10\phi +21 as \left(\phi ^{2}-7\phi \right)+\left(-3\phi +21\right).

$$\left(\phi ^{2}-7\phi \right)+\left(-3\phi +21\right)$$

Factor out \phi in the first and -3 in the second group.

$$\phi \left(\phi -7\right)-3\left(\phi -7\right)$$

Factor out common term \phi -7 by using distributive property.

$$\left(\phi -7\right)\left(\phi -3\right)$$

To find equation solutions, solve \phi -7=0 and \phi -3=0.

$$\phi =7 \phi =3$$

Steps Using the Quadratic Formula

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.

$$\phi ^{2}-10\phi +21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\phi =\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 21}}{2}$$

Square -10.

$$\phi =\frac{-\left(-10\right)±\sqrt{100-4\times 21}}{2}$$

Multiply -4 times 21.

$$\phi =\frac{-\left(-10\right)±\sqrt{100-84}}{2}$$

Add 100 to -84.

$$\phi =\frac{-\left(-10\right)±\sqrt{16}}{2}$$

Take the square root of 16.

$$\phi =\frac{-\left(-10\right)±4}{2}$$

The opposite of -10 is 10.

$$\phi =\frac{10±4}{2}$$

Now solve the equation \phi =\frac{10±4}{2} when ± is plus. Add 10 to 4.

$$\phi =\frac{14}{2}$$

Divide 14 by 2.

$$\phi =7$$

Now solve the equation \phi =\frac{10±4}{2} when ± is minus. Subtract 4 from 10.

$$\phi =\frac{6}{2}$$

Divide 6 by 2.

$$\phi =3$$

The equation is now solved.

$$\phi =7 \phi =3$$

Steps for Completing the Square

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.

$$\phi ^{2}-10\phi +21=0$$

Subtract 21 from both sides of the equation.

$$\phi ^{2}-10\phi +21-21=-21$$

Subtracting 21 from itself leaves 0.

$$\phi ^{2}-10\phi =-21$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\phi ^{2}-10\phi +\left(-5\right)^{2}=-21+\left(-5\right)^{2}$$

Square -5.

$$\phi ^{2}-10\phi +25=-21+25$$

Add -21 to 25.

$$\phi ^{2}-10\phi +25=4$$

Factor \phi ^{2}-10\phi +25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\phi -5\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\phi -5\right)^{2}}=\sqrt{4}$$

Simplify.

$$\phi -5=2 \phi -5=-2$$

Add 5 to both sides of the equation.

$$\phi =7 \phi =3$$