Forming Quadratic Equation from the Roots

When you're working with quadratic equations, sometimes you'll be given the roots (the solutions) and asked to form the equation itself. This might sound tricky, but don't worry! There are two simple methods you can use. Let's break them down step by step.

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What Are Roots?

Before we start, let's make sure we understand what roots are.

Roots are the solutions to a quadratic equation. If you have a quadratic equation like $x^2 - 2x - 3 = 0$, the roots are the values of $x$ that make the equation true. For example, if $x = 3$ and $x = -1$ are roots, then plugging these values into the equation should satisfy it (make it equal zero).

Method 1: Forming the Quadratic by Creating Factors

This method involves working backwards from the roots to form the equation.

Steps:

1. Let $x$ equal both of the roots.

• Suppose you know the roots are $r_1$ and $r_2$. We start by saying $x = r_1$ and $x = r_2$.
• For example, if the roots are $-1$ and $3$, we write: $x = -1 \quad \text{and} \quad x = 3$

2. Create two factors that equal zero.

• To turn each root into a factor, you need to move the root to the other side of the equation.
• For $x = -1$, move $-1$ to the other side to get $x + 1 = 0$.
• For $x = 3$, move $3$ to the other side to get $x - 3 = 0$.

3. Multiply the two factors together.

• Now, you have two factors: $(x + 1)$ and $(x - 3).$
• Multiply them together to form the quadratic equation: $(x + 1)(x - 3) = 0$
• Use the distributive property to expand: $x^2 - 3x + 1x - 3 = 0$
• Simplify the equation by combining like terms: $x^2 - 2x - 3 = 0$ Why it works:

When we move the roots to the other side of the equation, we're essentially "undoing" the solution process of the quadratic equation. Multiplying the factors gives us back the original equation.

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Method 2: Using the Formula

This method uses a specific formula that directly forms the quadratic equation from the roots. This method is a bit more mathematical, but it's very effective once you get the hang of it.

Formula:

$x^2 - (\text{sum of the roots})x + (\text{product of the roots}) = 0$

Steps:

1. Find the sum of the roots:

• Add the roots together.
• For example, if the roots are $-1$ and $3$: $\text{Sum of the roots} = (-1) + 3 = :highlight[2]$

2. Find the product of the roots:

• Multiply the roots together.
• For example: $\text{Product of the roots} = (-1) \times 3 = :highlight[-3]$

3. Substitute these values into the formula:

• Now, plug the sum and product into the formula: $x^2 - (2)x + (-3) = 0$
• Simplify the equation: $x^2 - 2x - 3 = 0$ Why it works:

This formula works because it directly builds the quadratic equation using the properties of roots. The sum of the roots gives you the middle term, and the product gives you the constant term.

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Which Method Should I Use?

• Method 1 (Creating Factors) is great if you like to see how the equation builds up from the roots. It's very hands-on and visual.
• Method 2 (Using the Formula) is quick and direct, especially useful if you're comfortable with substituting numbers into a formula. Both methods will give you the same result, so choose the one that makes the most sense to you. With practice, forming quadratic equations from their roots will become easier and more intuitive. Keep practicing, and you'll get the hang of it!