b) $$ x^{2}+9 x-36 $$

To factor the quadratic expression $$ x^{2} + 9x - 36 $$, follow these steps: ### Step 1: Identify the Coefficients The quadratic equation is in the form $$ ax^{2} + bx + c $$, where: - $$ a = 1 $$ - $$ b = 9 $$ - $$ c = -36 $$ ### Step 2: Find Two Numbers that Multiply to $$ a \times c $$ and Add to $$ b $$ We need to find two numbers that: - Multiply to $$ 1 \times (-36) = -36 $$ - Add up to $$ 9 $$ **Possible Pairs:** - $$ 12 $$ and $$ -3 $$ (since $$ 12 \times (-3) = -36 $$ and $$ 12 + (-3) = 9 $$) ### Step 3: Rewrite the Middle Term Using These Numbers Express the quadratic as: \[ x^{2} + 12x - 3x - 36 \] ### Step 4: Factor by Grouping Group the terms: \[ (x^{2} + 12x) + (-3x - 36) \] Factor out the greatest common factor from each group: \[ x(x + 12) - 3(x + 12) \] ### Step 5: Factor Out the Common Binomial \[ (x - 3)(x + 12) \] ### Final Factored Form \[ x^{2} + 9x - 36 = (x + 12)(x - 3) \] ### Verification (Optional) To ensure the factoring is correct, expand the factors: \[ (x + 12)(x - 3) = x(x) + x(-3) + 12(x) + 12(-3) = x^{2} - 3x + 12x - 36 = x^{2} + 9x - 36 \] Which matches the original expression. **Therefore, the factored form is:** \[ (x + 12)(x - 3) \] $$ x^{2} + 9x - 36 = (x + 12)(x - 3) $$

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b)

Ask by Hanson Lowe. in South Africa

Jan 21,2025

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The Deep Dive

To solve the quadratic equation

using the quadratic formula, you can substitute

, and

into the formula

. This yields two potential solutions:

and

. So, your solutions are

and

Did you know that the quadratic formula provides a universal method for solving quadratics, regardless of their complexity? It was popularized by the ancient Greeks and has been mathematically refined over centuries. So the next time you’re solving a quadratic, remember this hefty mathematical history behind that nifty formula!

In practice, quadratics are everywhere—from physics equations to business profit models. For instance, you can use quadratics to determine maximum profit in a given pricing strategy or to model the path of a ball thrown in the air. This real-world application shows just how integral these mathematical concepts are to everyday life!

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