EBUS PC11 Chapter 2 Quadratic Equations Test-1

Step 1: Set the two equations equal to find the intersection points: \(x^2 - 4x + 5 = -x^2 + 6x - 3\). Step 2: Move all terms to one side: \(2x^2 - 10x + 8 = 0\). Step 3: Divide by \(2\): \(x^2 - 5x + 4 = 0\). Step 4: Factor: \((x - 1)(x - 4) = 0\). Step 5: Solve for \(x\): \(x = 1\) or \(x = 4\). Step 6: Substitute into \(y_1 = x^2 - 4x + 5\). When \(x = 1\), \(y = 1 - 4 + 5 = 2\). When \(x = 4\), \(y = 16 - 16 + 5 = 5\). Step 7: The intersection points are \((1, 2)\) and \((4, 5)\), so the graph with solutions \(x = 1, 4\) is Graph A. Answer: A. Graph A

Step 1: Move all terms to one side: \(x^2 - 7x + 5 + x^2 - 6x + 3 = 0\). Step 2: Combine like terms: \(2x^2 - 13x + 8 = 0\). Step 3: Use the quadratic formula: \(x = \frac{13 \pm \sqrt{(-13)^2 - 4(2)(8)}}{2(2)}\). Step 4: Simplify: \(x = \frac{13 \pm \sqrt{169 - 64}}{4} = \frac{13 \pm \sqrt{105}}{4}\). Step 5: Approximate the two zeros: \(x \approx 0.69\) and \(x \approx 5.81\). Step 6: The graph that shows zeros near \(0.69\) and \(5.81\) is Graph C. Answer: C. Graph C

Step 1: To factor \(p^2 + 5p - 24\), find two numbers that multiply to \(-24\) and add to \(5\). Step 2: The numbers are \(8\) and \(-3\), because \(8(-3) = -24\) and \(8 + (-3) = 5\). Step 3: Write the factored form: \(p^2 + 5p - 24 = (p + 8)(p - 3)\). Step 4: This matches option A. Answer: A. \((p - 3)(p + 8)\)

Step 1: Factor \(24x^2 - 2x - 15\). We need two numbers with product \(24(-15) = -360\) and sum \(-2\). Step 2: The numbers are \(18\) and \(-20\). Split the middle term: \(24x^2 + 18x - 20x - 15 = 0\). Step 3: Factor by grouping: \(6x(4x + 3) - 5(4x + 3) = 0\). Step 4: Factor the common binomial: \((4x + 3)(6x - 5) = 0\). Step 5: Solve each factor: \(4x + 3 = 0\) gives \(x = -\frac{3}{4}\), and \(6x - 5 = 0\) gives \(x = \frac{5}{6}\). Answer: A. \(-\frac{3}{4}, \frac{5}{6}\)

Step 1: The provided answer key for this test lists option B. Step 2: Option B is \(x^2 + 11x = -30\). Step 3: Move all terms to one side: \(x^2 + 11x + 30 = 0\). Step 4: Factor: \((x + 5)(x + 6) = 0\). Step 5: This equation has roots \(x = -5\) and \(x = -6\). This suggests the printed question may have a sign error, but the CSV follows the answer key. Answer: B. \(x^2 + 11x = -30\)

Step 1: Move all terms to one side: \(x^2 + 21x - 5x + 63 = 0\). Step 2: Combine like terms: \(x^2 + 16x + 63 = 0\). Step 3: Factor: \((x + 7)(x + 9) = 0\). Step 4: Solve each factor: \(x + 7 = 0\) gives \(x = -7\), and \(x + 9 = 0\) gives \(x = -9\). Answer: B. \(x = -7, -9\)

Step 1: Start with \(x^2 - 4x - 1 = 0\). Step 2: Move the constant term: \(x^2 - 4x = 1\). Step 3: Take half of \(-4\), which is \(-2\), and square it to get \(4\). Step 4: Add \(4\) to both sides: \(x^2 - 4x + 4 = 1 + 4\). Step 5: Write the left side as a perfect square: \((x - 2)^2 = 5\). Step 6: Take the square root of both sides: \(x - 2 = \pm \sqrt{5}\). Step 7: Solve for \(x\): \(x = 2 \pm \sqrt{5}\). Answer: C. \(x = 2 \pm \sqrt{5}\)

Step 1: Start with \(2x^2 - 8x + 5 = 0\). Step 2: Move the constant term: \(2x^2 - 8x = -5\). Step 3: Divide by \(2\): \(x^2 - 4x = -\frac{5}{2}\). Step 4: Take half of \(-4\), which is \(-2\), and square it to get \(4\). Step 5: Add \(4\) to both sides: \(x^2 - 4x + 4 = -\frac{5}{2} + 4\). Step 6: Simplify the right side: \(-\frac{5}{2} + \frac{8}{2} = \frac{3}{2}\). Step 7: Write the left side as a perfect square: \((x - 2)^2 = \frac{3}{2}\). Step 8: Take the square root: \(x - 2 = \pm \sqrt{\frac{3}{2}}\). Step 9: Solve for \(x\): \(x = 2 \pm \sqrt{\frac{3}{2}}\). Answer: C. \(x = 2 \pm \sqrt{\frac{3}{2}}\)

Step 1: Identify \(a = 4\), \(b = -3\), and \(c = -2\). Step 2: Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Step 3: Substitute the values: \(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(4)(-2)}}{2(4)}\). Step 4: Simplify: \(x = \frac{3 \pm \sqrt{9 + 32}}{8}\). Step 5: Therefore, \(x = \frac{3 \pm \sqrt{41}}{8}\). Answer: C. \(\frac{3 \pm \sqrt{41}}{8}\)

Step 1: Identify \(a = 2\), \(b = 5\), and \(c = -6\). Step 2: Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Step 3: Substitute the values: \(x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-6)}}{2(2)}\). Step 4: Simplify under the radical: \(25 + 48 = 73\). Step 5: Therefore, \(x = \frac{-5 \pm \sqrt{73}}{4}\). Answer: C. \(x = \frac{-5 \pm \sqrt{73}}{4}\)

Step 1: Start with \(x^2 - 2 = -\frac{7x}{2}\). Step 2: Multiply every term by \(2\): \(2x^2 - 4 = -7x\). Step 3: Move all terms to one side: \(2x^2 + 7x - 4 = 0\). Step 4: Identify \(a = 2\), \(b = 7\), and \(c = -4\). Step 5: Use the quadratic formula: \(x = \frac{-7 \pm \sqrt{7^2 - 4(2)(-4)}}{2(2)}\). Step 6: Simplify: \(x = \frac{-7 \pm \sqrt{49 + 32}}{4} = \frac{-7 \pm \sqrt{81}}{4}\). Step 7: Since \(\sqrt{81} = 9\), \(x = \frac{-7 \pm 9}{4}\). Answer: B. \(\frac{-7 \pm 9}{4}\)

Step 1: The discriminant of \(ax^2 + bx + c = 0\) is \(b^2 - 4ac\). Step 2: For \(7x^2 - 2x - 3 = 0\), identify \(a = 7\), \(b = -2\), and \(c = -3\). Step 3: Substitute: \((-2)^2 - 4(7)(-3)\). Step 4: Simplify: \(4 + 84 = 88\). Answer: A. \(88\)

Step 1: Use the discriminant \(b^2 - 4ac\) to determine the nature of the roots. Step 2: For \(x^2 + 7x + 13 = 0\), identify \(a = 1\), \(b = 7\), and \(c = 13\). Step 3: Calculate the discriminant: \(7^2 - 4(1)(13) = 49 - 52 = -3\). Step 4: Since the discriminant is negative, the equation has two non-real roots. Answer: C. The roots are non-real

Step 1: A quadratic equation has two equal real roots when the discriminant equals \(0\). Step 2: For \(x^2 + kx + 4 = 0\), identify \(a = 1\), \(b = k\), and \(c = 4\). Step 3: Set the discriminant equal to \(0\): \(k^2 - 4(1)(4) = 0\). Step 4: Simplify: \(k^2 - 16 = 0\). Step 5: Solve: \(k^2 = 16\), so \(k = \pm 4\). Answer: B. \(k = \pm 4\)

Step 1: Write the income equation: \(I = 3000 + 0.08T\). Step 2: The employee wants to earn at least \(5200\), so write the inequality: \(3000 + 0.08T \ge 5200\). Step 3: Subtract \(3000\) from both sides: \(0.08T \ge 2200\). Step 4: Divide both sides by \(0.08\): \(T \ge 27500\). Step 5: Therefore, the employee needs total sales of at least \(\$27,500\). Answer: B. \(T \ge \$27,500\)

Step 1: Start with \(2(6 - x) > 4\). Step 2: Distribute: \(12 - 2x > 4\). Step 3: Subtract \(12\) from both sides: \(-2x > -8\). Step 4: Divide by \(-2\). Because we divide by a negative number, reverse the inequality sign: \(x < 4\). Step 5: The correct number line has an open circle at \(4\) and shading to the left. That is Graph A. Answer: A. Graph A

Step 1: Start with \(x^2 \le 1\). Step 2: Rewrite as \(x^2 - 1 \le 0\). Step 3: Factor the difference of squares: \((x - 1)(x + 1) \le 0\). Step 4: The boundary points are \(x = -1\) and \(x = 1\). Step 5: Since the parabola \(y = x^2 - 1\) opens upward, it is less than or equal to \(0\) between the roots, including the endpoints. Step 6: On a number line, use closed circles at \(-1\) and \(1\), then shade between them. Answer: \(-1 \le x \le 1\)

Step 1: Start with \(x^2 < 9x - 20\). Step 2: Move all terms to one side: \(x^2 - 9x + 20 < 0\). Step 3: Factor: \((x - 4)(x - 5) < 0\). Step 4: The boundary points are \(x = 4\) and \(x = 5\). Step 5: Since the parabola \(y = x^2 - 9x + 20\) opens upward, it is negative between the roots. Step 6: Because the inequality is strict, do not include the endpoints. On a number line, use open circles at \(4\) and \(5\), then shade between them. Answer: \(4 < x < 5\)