Roots of a Quadratic Equation

Correct answer: A. (-8)

Explanation: (\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta). From (52=36-2\alpha\beta), we get (\alpha\beta=-8).

Correct answer: A. 18

Explanation: Let the roots be \(t\) and \(2t\). Their sum is \(9\), so \(t+2t=9\), giving \(t=3\). Hence, the roots are \(3\) and \(6\). The product of the roots equals \(c\), so \(c=3\times6=18\). Exam tip: In \(x^2+bx+c=0\), the sum of the roots is \(-b\) and their product is \(c\).

Correct answer: A. 9

Explanation: Since both roots are negative and their ratio is \(1:2\), let them be \(-t\) and \(-2t\). Their product is \((-t)(-2t)=2t^2=18\), giving \(t=3\). Thus, the roots are \(-3\) and \(-6\), whose sum is \(-9\). By Vieta’s relation, the sum of the roots is \(-p\), so \(-p=-9\) and \(p=9\). Exam tip: for \(x^2+px+c=0\), the sum of the roots is \(-p\).

Correct answer: A. When \(c=a\)

Explanation: If the roots are \(\alpha\) and \(\beta\), then \(\alpha\beta=\frac{c}{a}\). Reciprocal roots have product \(1\), so \(c=a\). The condition \(b=0\) indicates additive inverse roots instead. Exam tip: use \(c/a\) for the product of roots.

Correct answer: A. (3)

Explanation: The sum is (-\frac{b}{a}=\frac{3}{4}) and the product is (\frac{k}{4}). From (\frac{k}{4}=\frac{3}{4}), (k=3).

Correct answer: A. \(m>0\)

Explanation: A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Here, \(a=1\), \(b=-4\), and \(c=m+4\), so \(D=b^2-4ac=16-4(m+4)=-4m\). Thus, \(-4m<0\), which gives \(m>0\). If \(m=0\), then \(D=0\), giving two equal real roots; therefore, option C is incorrect. Exam tip: For questions about the nature of roots, first determine the sign of the discriminant.

Correct answer: A. Every real \(m\)

Explanation: The equation can be rewritten as \((x-(m+2))^2=0\). Hence its repeated root is \(x=m+2\), and the roots are equal for every real value of \(m\). Therefore, option A is correct. Option C gives only the special case \(m=-2\), not the complete set of valid values. Exam tip: in such questions, first look for a perfect square or verify that the discriminant \(D=b^2-4ac\) is zero.

Correct answer: A. 4

Explanation: By Vieta’s formulas, \(\alpha+\beta=9\) and \(\alpha\beta=18\). Therefore, \((\alpha-2)(\beta-2)=\alpha\beta-2(\alpha+\beta)+4=18-18+4=4\). Hence, option A is correct. Exam tip: 18 is only the product \(\alpha\beta\); the required expression also involves the sum of the roots.

Correct answer: A. \(-12\)

Explanation: For a quadratic equation \(ax^2+bx+c=0\), the product of its roots is \(\frac{c}{a}\). Here, \(a=5\) and \(c=q\), so the product of the roots is \(\frac{q}{5}\). Therefore, \(\frac{q}{5}=3\times(-4)=-12\). The distractor \(12\) results from missing the negative sign. Exam tip: remember that the sum of roots is \(-\frac{b}{a}\), while their product is \(\frac{c}{a}\).

Correct answer: A. 7

Explanation: The sum of the roots is \(5+(-2)=3\). Since the sum of roots is \(-p\), we get \(p=-3\). Their product is \(q=5\times(-2)=-10\). Therefore, \(p-q=-3-(-10)=7\). Exam tip: For \(x^2+px+q=0\), the sum of roots is \(-p\) and their product is \(q\).

Correct answer: A. 32

Explanation: For the quadratic equation c(x^2-6x+2=0c), the sum of the roots is c(\alpha+\beta=6c) and their product is c(\alpha\beta=2c). Using the identity c(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\betac), we get c(\alpha^2+\beta^2=6^2-2(2)=36-4=32c). Therefore, option A is correct. Exam tip: use c(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\betac); 36 is only c((\alpha+\beta)^2c), not the required value.

Correct answer: A. (6)

Explanation: Here (\alpha+\beta=-2) and (\alpha\beta=-2). (\alpha^2+\alpha\beta+\beta^2=(\alpha+\beta)^2-\alpha\beta=6).

Correct answer: A. \(m+1\) and \(m+2\)

Explanation: If the roots are \(\alpha\) and \(\beta\), Vieta’s formulas give \(\alpha+\beta=2m+3\) and \(\alpha\beta=(m+1)(m+2)\). For option A, \((m+1)+(m+2)=2m+3\), and the product is exactly \((m+1)(m+2)\). Hence the roots are \(m+1\) and \(m+2\). Exam tip: compare the sum and product of the proposed roots with the coefficients of the quadratic.

Correct answer: A. 8

Explanation: By Vieta’s formulas, \(\alpha+\beta=10\) and \(\alpha\beta=24\). Therefore, \((\alpha-2)(\beta-2)=\alpha\beta-2(\alpha+\beta)+4=24-20+4=8\). Hence, option A is correct. Exam tip: when subtracting the same number from both roots, expand the product carefully; using only \(\alpha\beta\) would incorrectly give 24.

Correct answer: A. (1)

Explanation: The direct answer is option A: 1. To solve the equation x²−13x+42=0, look for two numbers with product 42 and sum 13. The numbers are 6 and 7, so x²−13x+42=(x−6)(x−7). Setting each factor equal to zero gives x=6 or x=7. Thus the smaller root is α=6 and the larger root is β=7. The required difference is β−α=7−6=1. Option A is correct because it states 1. Option B, 13, is the sum of the roots, 6+7=13, not their difference. Option C, 42, is the product of the roots, 6×7=42, not the difference. Option D, 6, is the smaller root itself, not the gap between the two roots. Always arrange the roots in the stated order before subtracting: larger minus smaller, not the reverse.

Correct answer: A. 144

Explanation: For a quadratic equation \(ax^2+bx+c=0\), the product of its roots is \(\alpha\beta=\frac{c}{a}\). Here, \(a=1\) and \(c=12\), so \(\alpha\beta=12\). Therefore, \(\alpha^2\beta^2=(\alpha\beta)^2=12^2=144\). Option 12 is only the value of \(\alpha\beta\), not its square. Exam tip: Remember that the sum and product of the roots are \(-\frac{b}{a}\) and \(\frac{c}{a}\), respectively.

Correct answer: A. 9 or -9

Explanation: The governing concept is the relationship between the roots and the coefficients of a quadratic equation, together with the fact that the labels α and β may be interchanged. Factoring gives x^2 - 9x + 20 = (x - 5)(x - 4), so the two roots are 5 and 4. Hence α + β = 5 + 4 = 9, which also follows from the coefficient rule α + β = -b/a = 9. If α = 5 and β = 4, then α - β = 1 and the required ratio is 9/1 = 9. If the labels are reversed, α = 4 and β = 5, then α - β = -1 and the ratio is 9/(-1) = -9. Therefore the possible values are 9 or -9, so option A is correct. Option B gives values that would require a different root difference, while options C and D do not follow from the actual sum and difference of the roots. The duplicate-option defect has been removed.

Correct answer: A. 7

Explanation: For a quadratic equation \(ax^2+bx+c=0\), the sum of the roots is \(\alpha+\beta=-\frac{b}{a}\) and their product is \(\alpha\beta=\frac{c}{a}\). Thus, here \(\alpha+\beta=4\) and \(\alpha\beta=-5\). Therefore, \((\alpha+2)(\beta+2)=\alpha\beta+2(\alpha+\beta)+4=-5+2(4)+4=7\). Hence, option A is correct. Exam tip: Use the sum and product of roots directly instead of solving for the roots individually.

Correct answer: A. 14

Explanation: Let the equal root be \(r\). Since the product of the roots is \(49\), we have \(r^2=49\), giving \(r=\pm7\). Because the roots are negative, \(r=-7\). Thus, their sum is \(-14\). By Vieta’s formula, the sum of the roots is \(-k\), so \(-k=-14\) and \(k=14\). Exam tip: use the constant term to find the equal root first, then use the sum of roots \(-k\) to determine the coefficient.

Correct answer: A. (\frac{106}{45})

Explanation: Here (\alpha+\beta=14) and (\alpha\beta=45). (\alpha^2+\beta^2=196-90=106), so the value is (\frac{106}{45}).

Correct answer: A. \(x^2-6x+9=0\)

Explanation: For \(ax^2+bx+c=0\), roots are real and equal when the discriminant \(D=b^2-4ac\) is zero. In option A, \(D=(-6)^2-4(1)(9)=36-36=0\). In option B, \(D=-4\), so its roots are not real. Exam tip: check \(D=0\) first for equal roots.

Correct answer: A. (x^2-\frac{7}{10}x+\frac{1}{10}=0)

Explanation: The old sum is (7) and product is (10). The reciprocal roots have sum (\frac{7}{10}) and product (\frac{1}{10}).

Correct answer: A. (20)

Explanation: Here (\alpha+\beta=5) and (\alpha\beta=5). (\alpha^2+\beta^2+\alpha\beta=(\alpha+\beta)^2-\alpha\beta=20).

Correct answer: A. -24

Explanation: By Vieta’s formulas, \(\alpha+\beta=3\) and \(\alpha\beta=-28\). Therefore, \((\alpha+1)(\beta+1)=\alpha\beta+\alpha+\beta+1=-28+3+1=-24\). Hence, option A is correct. Remember to include the final \(+1\) term when expanding the product; omitting it leads to an incorrect result.

Correct answer: A. (\frac{13}{36})

Explanation: Here (\alpha+\beta=11) and (\alpha\beta=24). The value is (\frac{\alpha+\beta+2}{\alpha\beta+\alpha+\beta+1}=\frac{13}{36}).