\[ (x + 5)(x - 3) = 120 \] - United Radiology

["# How to Solve the Equation ((x + 5)(x - 3) = 120): Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, and one common equation students often encounter is ((x + 5)(x - 3) = 120). Whether you’re preparing for exams or tackling math homework, understanding how to solve this equation efficiently can save time and improve problem-solving confidence.", "In this article, we’ll explore multiple approaches to solving ((x + 5)(x - 3) = 120), including expanding, factoring, and completing the square, along with practical examples to help you grasp each method clearly.", "---", "## Understanding the Equation", "The equation ((x + 5)(x - 3) = 120) is a quadratic equation in factored form. Expanding it reveals a standard quadratic expression, which allows us to apply familiar solving techniques like the quadratic formula or factoring.", "### Step 1: Expand the Left Side", "[
\n(x + 5)(x - 3) = x^2 - 3x + 5x - 15 = x^2 + 2x - 15
\n]", "So the equation becomes:", "[
\nx^2 + 2x - 15 = 120
\n]", "### Step 2: Move All Terms to One Side", "Subtract 120 from both sides to form a standard quadratic equation:", "[
\nx^2 + 2x - 135 = 0
\n]", "Now the equation is:", "[
\nx^2 + 2x - 135 = 0
\n]", "---", "## Solving the Quadratic Equation", "### Method 1: Factoring", "Look for two numbers that multiply to (-135) and add to (2).", "After testing factor pairs of (-135), we find:", "[
\nx + 15 \quad \ ext{and} \quad x - 9 \quad \ ext{since} \quad (x + 15)(x - 9) = x^2 + 6x - 135 \quad \ ext{(incorrect sum)}
\n]", "Wait — correction: correct factors of (-135) that add to (+2):", "Try (13) and (-11): no. Try (15) and (-9):", "[
\n(x + 15)(x - 9) = x^2 + 6x - 135 \quad \ ext{incorrect sum}
\n]", "Wait again — correct pairing:
\nFind (a) and (b) such that:
\n(a \cdot b = -135) and (a + b = 2)", "After checking:", "[
\n13 \ imes (-11) = -143 \quad \ ext{no}
\n]", "(15 \ imes (-9) = -135), and (15 - 9 = 6) — no.
\nTry (13) and (-11)? No.", "Actually, correct factorization:", "Try (13) and (-11) no. Wait — test systematically:", "Factors of (-135):", "- (135 = 1 \ imes 135), so try:", "(x = 15): ((15 + 5)(15 - 3) = 20 \ imes 12 = 240) ≠ 120
\nTry (x = 9): ((14)(6) = 84) ≠ 120
\nBut later we solve algebraically — factoring may not be intuitive immediately.", "Instead, use the correct factored form after expansion:", "We already have:
\n[
\nx^2 + 2x - 135 = 0
\n]", "Now use the quadratic formula, since factoring is not straightforward here:", "[
\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "With (a = 1), (b = 2), (c = -135):", "[
\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-135)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 540}}{2} = \frac{-2 \pm \sqrt{544}}{2}
\n]", "Simplify (\sqrt{544}):", "[
\n544 = 16 \ imes 34 \Rightarrow \sqrt{544} = \sqrt{16 \ imes 34} = 4\sqrt{34}
\n]", "Thus:", "[
\nx = \frac{-2 \pm 4\sqrt{34}}{2} = -1 \pm 2\sqrt{34}
\n]", "---", "### Method 2: Completing the Square", "Start from:", "[
\nx^2 + 2x - 135 = 0
\n]", "Move constant to the right:", "[
\nx^2 + 2x = 135
\n]", "Take half of 2 → 1, square it → 1, add to both sides:", "[
\nx^2 + 2x + 1 = 135 + 1 \Rightarrow (x + 1)^2 = 136
\n]", "Take square roots:", "[
\nx + 1 = \pm \sqrt{136} = \pm \sqrt{4 \ imes 34} = \pm 2\sqrt{34}
\n]", "So:", "[
\nx = -1 \pm 2\sqrt{34}
\n]", "Same result as before.", "---", "## Verifying the Solutions", "Use (x = -1 + 2\sqrt{34}):", "[
\nx + 5 = 4 + 2\sqrt{34},\quad x - 3 = 1 + 2\sqrt{34}
\n]", "Multiply:", "[
\n(x + 5)(x - 3) = (4 + 2\sqrt{34})(1 + 2\sqrt{34}) = 4(1) + 4(2\sqrt{34}) + 2\sqrt{34}(1) + 4(\sqrt{34})^2
\n]
\n[
\n= 4 + 8\sqrt{34} + 2\sqrt{34} + 4 \ imes 34 = 4 + 10\sqrt{34} + 136 = 140 + 10\sqrt{34}
\n]", "Wait — this does not equal 120. What happened?", "Ah! Here’s the mistake: completing the square gave ((x + 1)^2 = 136), so (x + 1 = \pm \sqrt{136}), so:", "[
\nx = -1 \pm \sqrt{136} = -1 \pm 2\sqrt{34}
\n]", "Now recompute product:", "Let (x = -1 + 2\sqrt{34}), then:", "[
\nx + 5 = 4 + 2\sqrt{34},\quad x - 3 = 1 + 2\sqrt{34}
\n]", "Now compute:", "[
\n(x + 5)(x - 3) = (4 + 2\sqrt{34})(1 + 2\sqrt{34}) = 4(1) + 4(2\sqrt{34}) + 2\sqrt{34}(1) + (2\sqrt{34})(2\sqrt{34})
\n]
\n[
\n= 4 + 8\sqrt{34} + 2\sqrt{34} + 4 \ imes 34 = 4 + 10\sqrt{34} + 136 = 140 + 10\sqrt{34}
\n]", "But this contradicts the original equation being 120. That means our earlier expansion is correct — but why?", "Wait — let's double-check the expansion:", "[
\n(x + 5)(x - 3) = x^2 - 3x + 5x - 15 = x^2 + 2x - 15
\n]", "Set equal to 120:", "[
\nx^2 + 2x - 135 = 0
\n]", "So solutions are indeed:", "[
\nx = \frac{-2 \pm \sqrt{4 + 540}}{2} = \frac{-2 \pm \sqrt{544}}{2} = \frac{-2 \pm 4\sqrt{34}}{2} = -1 \pm 2\sqrt{34}
\n]", "Now, check:", "Let’s compute ((x + 5)(x - 3)) with (x = -1 + 2\sqrt{34} \approx -1 + 2 \ imes 5.83 = -1 + 11.66 = 10.66)", "Then:", "[
\n(x + 5) \approx 15.66,\quad (x - 3) \approx 7.66,\quad \ ext{product} \approx 120
\n]", "Yes — approximately correct. So exact form is:", "[
\nx = -1 \pm 2\sqrt{34}
\n]", "---", "## Why Factor Mentions ‘(x + 5)(x - 3) = 120’ — Common Mistake Avoided", "Many students expect to factor ((x + 5)(x - 3)) directly, but it’s not factorable over integers. Instead, they should expand first or recognize it's already factored and proceed with substitution.", "---", "## Final Answer", "The solutions to the equation ((x + 5)(x - 3) = 120) are:", "[
\n\boxed{x = -1 + 2\sqrt{34} \quad} \ ext{and} \quad \boxed{x = -1 - 2\sqrt{34}}
\n]", "---", "## Bonus Tips: Graphical Interpretation", "The left-hand side, ((x + 5)(x - 3)), is a parabola opening upwards with roots at (x = -5) and (x = 3). The equation ((x + 5)(x - 3) = 120) checks where this parabola intersects the horizontal line (y = 120). The quadratic we derived confirms two real solutions, meaning the parabola crosses (y = 120) at two points — matching our solutions.", "---", "## Why This Equation Matters", "Mastering such equations strengthens algebraic reasoning, prepares students for more advanced topics (like systems of equations or quadratic optimization), and improves problem-solving speed in standardized tests.", "---", "## Frequently Asked Questions (FAQ)", "Q: Can I solve ((x + 5)(x - 3) = 120) quickly without expanding?
\nA: Yes, expand to get (x^2 + 2x - 135 = 0), then apply quadratic formula or completing the square — both are valid and efficient.", "Q: What if the equation doesn’t factor nicely?
\nA: Use the quadratic formula — it always works for any real quadratic.", "Q: How accurate are the decimal approximations?
\nA: They verify the correctness but should not replace exact forms like (-1 \pm 2\sqrt{34}).", "---", "### Conclusion", "Solving ((x + 5)(x - 3) = 120) involves expanding, forming a quadratic, and applying standard solving techniques — either factoring (after expansion), completing the square, or the quadratic formula. Understanding each step builds confidence and deepens algebraic mastery.", "Keep practicing — math becomes easier with persistence!"]