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**problem**

Solve $\color{blue}{x^2-8x+15 = 0}$ using factoring.

**solution**

**The solutions are:**

**explanation**

First we need to factor trinomial $ \color{blue}{ x^2-8x+15 } $ and than we use factored form to solve an equation $ \color{blue}{ x^2-8x+15 = 0} $.

** Step 1:** Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$.
( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

Now we must discover two numbers that sum up to $ \color{blue}{ -8 } $ and multiply to $ \color{red}{ 15 } $.

** Step 2:** Find out pairs of numbers with a product of $\color{red}{ c = 15 }$.

PRODUCT = 15 | |

1 15 | -1 -15 |

3 5 | -3 -5 |

** Step 3:** Find out which pair sums up to $\color{blue}{ b = -8 }$

PRODUCT = 15 and SUM = -8 | |

1 15 | -1 -15 |

3 5 | -3 -5 |

** Step 4:** Put -3 and -5 into placeholders to get factored form.

** Step 5:** Set each factor to zero and solve equations.

working...

EXAMPLES

example 1:ex 1:

Solve for $x^2 + 3x - 4 = 0$ by factoring.

example 2:ex 2:

Solve $4x^2 - x - 3 = 0$ by completing the square.

example 3:ex 3:

Solve $-2x^2 - 0.5x + 0.75 = 0$ using the quadratic formula.

example 4:ex 4:

Solve $ \frac{2}{3} x^2 - \frac{1}{3} x - 5 = 0 $.

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TUTORIAL

The most commonly used methods for solving quadratic equations are:

**1**. Factoring method

**2**. Completing the square

**3**. Using quadratic formula

In the following sections, we'll go over these methods.

**If** a quadratic trinomial can be factored, this is the best solving method.

We often use this method when the leading coefficient is equal to 1 or -1. If this is not the case, then it is better to use some other method.

**Example 01:** Solve $ x^2 \color{red}{-8}x \color{blue}{+ 15} = 0 $ by factoring.

Here we see that the leading coefficient is 1, so the factoring method is our first choice.

To factor this equation, we must find two numbers ( $ a $ and $ b $ ) with a sum is $ a + b = \color{red}{8} $ and a product of $ a \cdot b = \color{blue}{15} $.

After some trials and errors, we see that $ a = 3 $ and $ b = 5 $.

Now we use formula $ x^2 - 8x + 15 = (x - a)(x - b) $ to get factored form:

$$ x^2 - 8x + 15 = (x - 3)(x - 5) $$Divide the factored form into two linear equations to get solutions.

$$ \begin{aligned} x^2 - 8x + 15 &= 0 \\ (x - 3)(x - 5) &= 0 \\ x -3 &= 0 ~~ \text{or} ~~ x - 5= 0 \\ x &= 3 ~~ \text{or} ~~ x = 5 \end{aligned} $$**Example 02:** Solve $ x^2 -8x = 0 $ by factoring.

In this case, (when the coefficient **c = 0** ) we can factor out $ \color{blue}{x} $ out of $ x^2 - 8x $.

**Example 03:** Solve $ x^2 - 16 = 0 $ by factoring.

In this case, ( when the middle term is equal 0) we can use the difference of squares formula.

$$ \begin{aligned} x^2 - 16 &= 0 \\ x^2 - 4^2 &= 0 \text{ use } a^2 - b^2 = (a-b)(a+b) \\ (x - 4)(x+4) &= 0 \\ x - 4 &= 0 ~~ \text{or} ~~ x + 4 = 0 \\ x &= 4 ~~ \text{or} ~~ x = -4 \end{aligned} $$This method solves all types of quadratic equations. It works best when solutions contain some radicals or complex numbers.

**Example 05:** Solve equation $ 2x^2 + 3x - 2 = 0$ by using quadratic formula.

**Step 1**: Read the values of $a$, $b$, and $c$ from the quadratic equation.
( $a$ is the number in front of $x^2$ , $b$ is the number in front of $x$ and $c$ is the number at the end)

**Step 2**:Plug the values for a, b, and c into the quadratic formula and simplify.

**Step 3**: Solve for $x_1$ and $x_2$

This method can be used to solve all types of quadratic equations, although it can be complicated for some types of equations. The method involves seven steps.

**Example 04:** Solve equation $ 2x^2 + 8x - 10= 0$ by completing the square.

**Step 1**: Divide the equation by the number in front of the square term.

**Step 2**: move $-5$ to the right:

**Step 3**: Take half of the x-term coefficient $ \color{blue}{\dfrac{4}{2}} $, square it
$ \color{blue}{\left(\dfrac{4}{2} \right)^2} $ and add this value to both sides.

**Step 4**: Simplify left and right side.

**Step 5**: Write the perfect square on the left.

**Step 6**: Take the square root of both sides.

**Step 7**: Solve for $x_1$ and $x_2$ .

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