1.
Question 1. One of the factors of \(x^2+4x-12\) is
Step 1: Find two numbers that multiply to \(-12\) and add to \(4\). Step 2: The numbers are \(6\) and \(-2\). Step 3: Factor: \(x^2+4x-12=(x+6)(x-2)\). Answer: \((x+6)\)
2.
Question 2. Factor: \(3x^2-17x+10\)
Step 1: Multiply \(3\cdot10=30\). Step 2: Find two numbers that multiply to \(30\) and add to \(-17\): \(-15\) and \(-2\). Step 3: Split: \(3x^2-15x-2x+10\). Step 4: Factor by grouping: \(3x(x-5)-2(x-5)\). Step 5: Factor: \((3x-2)(x-5)\). Answer: \((3x-2)(x-5)\)
3.
Question 3. How does the graph of a quadratic function in the form \(y=a(x-p)^2+q\) change when the value of \(p\) is decreased by \(3\), the value of \(q\) is increased by \(2\), and the sign of \(a\) is changed to its opposite?
Step 1: In vertex form, \(p\) controls horizontal movement. Step 2: Decreasing \(p\) by \(3\) moves the graph \(3\) units left. Step 3: Increasing \(q\) by \(2\) moves the graph \(2\) units up. Step 4: Changing the sign of \(a\) reflects the graph in the \(x\)-axis. Answer: The graph is reflected in the x-axis, and translated 3 units left and 2 units up.
4.
Question 4. The graph of \(y=5x^2-6x-8\) is shown below. What are the zeros of the function as exact values, the y-intercept, and the maximum or minimum value of the function rounded to the nearest tenth?
Step 1: Factor: \(5x^2-6x-8=(5x+4)(x-2)\). Step 2: Set each factor to zero to get \(x=-\frac{4}{5}\) and \(x=2\). Step 3: The y-intercept is found at \(x=0\): \(y=-8\), so \((0,-8)\). Step 4: The vertex x-coordinate is \(x=-\frac{b}{2a}=\frac{6}{10}=0.6\). Step 5: Substitute: \(5(0.6)^2-6(0.6)-8=-9.8\). Step 6: Since \(a>0\), this is a minimum. Answer: Zeros: \(x=-\frac{4}{5}, 2\); y-intercept: \((0,-8)\); Minimum value of \(-9.8\).
5.
Question 5. What is the equation of the parabola with vertex \((-5,-1)\), that opens up, and is congruent to \(y=-3x^2\)?
Step 1: Vertex form is \(y=a(x-h)^2+k\). Step 2: The vertex \((-5,-1)\) gives \(h=-5\) and \(k=-1\), so \(y=a(x+5)^2-1\). Step 3: Congruent to \(y=-3x^2\) means \(|a|=3\). Step 4: The parabola opens up, so \(a=3\). Answer: \(y=3(x+5)^2-1\)
6.
Question 6. When the quadratic equation \(y=-x^2-4x-3\) is rewritten in the form \(y=a(x-p)^2+q\), what is the value of \(p\)?
Step 1: Factor \(-1\) from the quadratic and linear terms: \(y=-(x^2+4x)-3\). Step 2: Complete the square: \(x^2+4x=(x+2)^2-4\). Step 3: Substitute: \(y=-((x+2)^2-4)-3\). Step 4: Simplify: \(y=-(x+2)^2+1\). Step 5: Since \((x+2)^2=(x-(-2))^2\), \(p=-2\). Answer: \(-2\)
7.
Question 7. A company can sell \(2000\) magazine subscriptions at \($40\) dollars each. For each \($5\) increase in the price, it will sell \(200\) fewer subscriptions. What subscription price will provide the maximum revenue for the company?
Step 1: Let \(x\) be the number of \($5\) price increases. Step 2: Price is \(40+5x\), and subscriptions sold are \(2000-200x\). Step 3: Revenue is \(R=(40+5x)(2000-200x)\). Step 4: Expand: \(R=-1000x^2+2000x+80000\). Step 5: The maximum occurs at \(x=-\frac{b}{2a}=-\frac{2000}{2(-1000)}=1\). Step 6: Price is \(40+5(1)=45\). Answer: \($45\)
8.
Question 8. What is the product of the roots of \(18x^2+45x+7=0\)?
Step 1: For \(ax^2+bx+c=0\), the product of the roots is \(\frac{c}{a}\). Step 2: Here \(a=18\) and \(c=7\). Step 3: The product is \(\frac{7}{18}\). Answer: \(\frac{7}{18}\)
9.
Question 9. When the quadratic function \(4x^2+24x=11\) is solved by completing the square, the exact solutions can be written in the form \(x=-3\pm\frac{\sqrt{A}}{B}\). The value of \(A\) is
Step 1: Divide by \(4\): \(x^2+6x=\frac{11}{4}\). Step 2: Complete the square by adding \(9\) to both sides: \(x^2+6x+9=\frac{11}{4}+9\). Step 3: Simplify: \((x+3)^2=\frac{47}{4}\). Step 4: Take the square root: \(x+3=\pm\frac{\sqrt{47}}{2}\). Step 5: Thus \(A=47\). Answer: \(47\)
10.
Question 10. Solve \(\frac{4x^2}{3}=4x-2\) by using the Quadratic Formula. The roots can be expressed as \(x=\frac{R\pm\sqrt{S}}{T}\). What is the value of \(S\)?
Step 1: Multiply both sides by \(3\): \(4x^2=12x-6\). Step 2: Move all terms to one side: \(4x^2-12x+6=0\). Step 3: Use the quadratic formula with \(a=4\), \(b=-12\), \(c=6\). Step 4: The discriminant is \((-12)^2-4(4)(6)=144-96=48\). Step 5: \(\sqrt{48}=4\sqrt{3}\). Step 6: Simplifying gives \(x=\frac{3\pm\sqrt{3}}{2}\), so \(S=3\). Answer: \(3\)
11.
Question 11. For what values of \(k\) does \(x^2+kx+7=0\) have no real roots?
Step 1: A quadratic has no real roots when the discriminant is less than zero. Step 2: For \(x^2+kx+7=0\), the discriminant is \(k^2-4(1)(7)=k^2-28\). Step 3: Set \(k^2-28<0\). Step 4: Then \(k^2<28\). Step 5: Therefore \(-\sqrt{28}<k<\sqrt{28}\). Step 6: Simplify \(\sqrt{28}=2\sqrt{7}\). Answer: \(-2\sqrt{7}<k<2\sqrt{7}\)
12.
Question 12. Eight less than the product of \(-3\) and a number, \(x\), is greater than or equal to \(-26\). The solution to this inequality is _______.
Step 1: The product of \(-3\) and \(x\) is \(-3x\). Step 2: Eight less than that is \(-3x-8\). Step 3: Write the inequality: \(-3x-8\geq-26\). Step 4: Add \(8\): \(-3x\geq-18\). Step 5: Divide by \(-3\) and reverse the inequality: \(x\leq6\). Answer: \(x\leq6\)
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