Introduction to Quadratic Equations

Correct answer: A. (9x^2-4x-42=0)

Multiplying the whole equation by (12) gives (9x^2+6-4x+20=72). Therefore the standard form is (9x^2-4x-46=0).

Correct answer: A. 11

Since \(x=-5\) is a root, substitute it directly into the equation: \(3(-5)^2+p(-5)-20=0\). Thus, \(75-5p-20=0\), so \(55-5p=0\) and \(p=11\). The distractor \(-11\) results from mishandling the negative sign in \(p(-5)\). Exam tip: substitute the given root carefully and simplify before solving for the parameter.

Correct answer: A. (x^2+13x+42=0)

A monic equation is (x^2-(\text{sum})x+\text{product}=0). Substituting sum (-13) gives (x^2+13x+42=0).

Correct answer: A. \(m=\pm7\)

For equal roots, the discriminant \(D=b^2-4ac\) must be zero. Here \(a=1, b=-2m, c=49\), so \(D=(-2m)^2-4(1)(49)=4m^2-196=0\). Thus, \(m^2=49\), giving \(m=\pm7\). Choosing only \(m=7\) or only \(m=-7\) omits one valid value. Exam tip: For equal roots of a quadratic equation, set the discriminant directly equal to zero.

Correct answer: A. \(p\leq -6\) या \(p\geq 6\)

A quadratic equation has real roots when its discriminant satisfies \(D\geq 0\). Here, \(a=1, b=2p, c=36\), so \(D=(2p)^2-4(1)(36)=4p^2-144\). Thus, \(4p^2-144\geq 0\), which gives \(p^2\geq 36\) and hence \(p\leq -6\) or \(p\geq 6\). In option B, the discriminant is negative, so the roots are not real. Exam tip: include \(D=0\), because equal roots are also real roots.

Correct answer: A. ( \frac{17}{6} )

The sum of reciprocals is (\frac{\alpha+\beta}{\alpha\beta}). Here it is (\frac{\frac{17}{5}}{\frac{6}{5}}=\frac{17}{6}).

Correct answer: A. 71

By Vieta’s formulas, the sum of the roots is \(s=7+8=15\), and their product is \(p=7\times8=56\). Therefore, \(s+p=15+56=71\), so option A is correct. Exam tip: In an equation of the form \(x^2-sx+p=0\), the sum of the roots is \(s\) and their product is \(p\).

Correct answer: A. \(k<64\)

A quadratic equation has real and distinct roots only when its discriminant is positive. Here, \(D=(-16)^2-4(1)(k)=256-4k\). Thus, \(256-4k>0\), which gives \(k<64\). At \(k=64\), the discriminant is zero and the roots are equal, so option B is not correct. Exam tip: For real and distinct roots, always apply the condition \(D>0\).

Correct answer: A. x² + x + 9 = 0

For a quadratic equation ax² + bx + c = 0, the discriminant is D = b² − 4ac. In option A, a = 1, b = 1, and c = 9, so D = 1² − 4 × 1 × 9 = 1 − 36 = −35. For example, option C gives 2² − 4 × 3 × 3 = −32, so it is not correct. Exam tip: the sign of b disappears when calculating b², but the complete value of 4ac must be subtracted.

Correct answer: A. ( \frac{5}{2} )

The direct answer is option A, \(5/2\). Let the roots of \(4x^2+8x+3=0\) be \(\alpha\) and \(\beta\). Using the root relations, \(\alpha+\beta=-8/4=-2\) and \(\alpha\beta=3/4\). The identity for the sum of squares is \(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta\). Substitution gives \((-2)^2-2(3/4)=4-3/2=5/2\). Hence option A is correct. Option B, 4, is only the square of the sum and forgets the subtraction of \(2\alpha\beta\). Option C, \(3/4\), is the product of the roots, not the sum of their squares. Option D, \(13/4\), results from an incorrect operation and does not match the identity. The roots themselves need not be solved. Memory cue: square-sum = square of sum − twice product.

Correct answer: A. 0

By Vieta’s formulas, \(\alpha+\beta=14\) and \(\alpha\beta=48\). Therefore, \((\alpha-6)(\beta-6)=\alpha\beta-6(\alpha+\beta)+36=48-6(14)+36=0\). Hence, option A is correct. Option C is only the value of \(\alpha\beta\), not of the complete expression. Exam tip: In such questions, use the sum and product of the roots instead of finding the roots separately.

Correct answer: A. 2x²+14x−78=0

Expand the two products: (x+4)(x+7)=x²+11x+28 and (x−3)(x+6)=x²+3x−18. Their sum is 2x²+14x+10, so the equation becomes 2x²+14x+10=88. Bringing 88 to the left gives 2x²+14x−78=0, so option A is correct. Option B stops before subtracting 88. Exam tip: To write a quadratic equation in standard form, bring all terms to one side so that the other side is 0.

Correct answer: A. (90)

The direct answer is option A: 90. A monic quadratic has leading coefficient 1, so if its roots are r₁ and r₂, its equation is x² - (r₁ + r₂)x + r₁r₂ = 0. Therefore the constant term equals the product of the roots. The roots here are 2r and 5r, and their sum is 21. Thus 2r + 5r = 21, so 7r = 21 and r = 3. The roots are therefore 2(3) = 6 and 5(3) = 15. Their product is 6 × 15 = 90, so the constant term is 90. Option A is correct. Option B, 21, is the sum of the roots, not their product. Option C, 45, does not equal the product 6 × 15 and may result from an incomplete calculation. Option D, 63, is another incorrect product or arithmetic value; it is not the constant term. The word “monic” is important: because the leading coefficient is 1, the constant term directly equals the root product. Memory cue: first find r from the sum, then multiply the two roots.

Correct answer: A. It has no real roots

Here, \(a=1\), \(b=-10\), and \(c=29\). The discriminant is \(D=b^2-4ac=(-10)^2-4(1)(29)=100-116=-16\). Since \(D<0\), the equation has no real roots; its roots are complex. Option B would be correct only if \(D=0\), which is not the case here. Exam tip: For a quadratic equation, \(D<0\) always indicates that there are no real roots.

Correct answer: A. \(x(x-3)=4\)

Expanding \(x(x-3)=4\) gives \(x^2-3x-4=0\), whose highest power of \(x\) is 2, so it is quadratic. Option B is an identity because both sides are identical. Exam tip: bring all terms to one side, then check the highest power.

Correct answer: A. \(\frac{2}{5}\)

Factoring gives \(5x^2-22x+24=(5x-12)(x-2)\). Therefore, the roots are \(\frac{12}{5}\) and \(2\). The difference between the larger and smaller roots is \(\frac{12}{5}-2=\frac{2}{5}\). The values \(\frac{12}{5}\) and \(2\) are the roots themselves, not their difference. In an exam, subtract the smaller root from the larger root when the difference between roots is asked.

Correct answer: A. 81

For the expression to be a perfect square, \(x^2-18x+c\) must equal \((x-9)^2=x^2-18x+81\). Therefore, \(c=81\). Choosing 324 incorrectly amounts to squaring 18 directly; the correct method is to square half the coefficient of \(x\). Exam tip: \(x^2+bx+c\) is a perfect square when \(c=\left(\frac{b}{2}\right)^2\).

Correct answer: A. 20

By Vieta’s formula, the sum of the roots of \(x^2+bx+100=0\) is \(-b\). Since both roots are \(-10\), their sum is \(-10+(-10)=-20\). Therefore, \(-b=-20\), giving \(b=20\). Exam tip: for \(x^2+bx+c=0\), the sum of the roots is always \(-b\); missing this negative sign leads to option B.

Correct answer: B. \(a>0,\ b>0,\ c<0\)

For \(ax^2+bx+c=0\), sum of roots is \(\alpha+\beta=\frac{-b}{a}\) and product of roots is \(\alpha\beta=\frac{c}{a}\). We need \(\alpha+\beta<0\Rightarrow \frac{-b}{a}<0\Rightarrow \frac{b}{a}>0\), so \(a\) and \(b\) must have the same sign. Also \(\alpha\beta<0\Rightarrow \frac{c}{a}<0\), so \(a\) and \(c\) must have opposite signs. The only option with \(a\) and \(b\) same sign and \(c\) opposite is \(a>0,\ b>0,\ c<0\).

Correct answer: A. \(x^2+3x-184=0\)

Area of a rectangle = length × breadth, so \((x+9)(x-6)=130\). Expanding gives \(x^2+3x-54=130\), and rearranging produces \(x^2+3x-184=0\). Option B does not correctly account for the \(-54\) term, while options C and D contain errors in expansion. Exam tip: multiply the binomials first, then bring all terms to one side and equate the expression to zero.

Correct answer: A. (x^2-17x+72=0)

If the sum of roots is (17) and product is (72), the equation is (x^2-17x+72=0). Remember the monic form formula.

Correct answer: A. ( \frac{13}{42} )

(\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}). Here the value is (\frac{13}{42}).

Correct answer: B. \(-10\)

Expand \((x+a)^2\): \((x+a)^2=x^2+2ax+a^2\).
Compare with \(x^2-20x+100\):
So \(2a=-20\Rightarrow a=-10\). Also \(a^2=100\) matches. Hence \(a=-10\).

Correct answer: A. Two equal real roots

The equation can be written as 25x² + 40kx + 16k² = (5x + 4k)². Hence, (5x + 4k)² = 0 gives x = −4k/5, which is a repeated root. Equivalently, the discriminant is D = (40k)² − 4(25)(16k²) = 0, so the roots are equal and real. Option B is incorrect because distinct real roots require D > 0. Exam tip: For a quadratic equation, D = 0 indicates two equal real roots.

Correct answer: A. It is not possible

If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (49), so it is not possible.