Q: What is the nature of the roots of the quadratic equation $16x^2-24kx+9k^2=0$?

Two equal real roots The equation can be written as $(4x-3k)^2=0$. Hence $4x-3k=0$, giving the repeated root $x=\frac{3k}{4}$. Equivalently, the discriminant is $D=b^2-4ac=0$, which confirms that the roots are equal and real. Exam tip: for a quadratic equation, $D=0$ indicates two equal real roots.

Q: If (x=2) is not a root of (kx^2-7x+3=0), what condition must (k) satisfy?

(k\neq \frac{11}{4}) Putting (x=2), the left side becomes (4k-14+3=4k-11). For it not to be a root, (4k-11\neq0), so (k\neq\frac{11}{4}).

Q: Which quadratic equation has sum of roots (0) and product (-121)?

(x^2-121=0) The monic equation is (x^2-(\text{sum})x+\text{product}=0). Using sum (0) and product (-121) gives (x^2-121=0).

Q: If in $4x^2+kx+28=0$, the product of roots is (-2) times the sum of roots, what is (k)?

(14) The direct answer is option A, $k=14$. For $4x^2+kx+28=0$, let the roots be $\alpha$ and $\beta$. By the relations between coefficients and roots, their sum is $\alpha+\beta=-k/4$, and their product is $\alpha\beta=28/4=7$. The question says the product is −2 times the sum, so $7=-2(-k/4)$. The right side simplifies to $k/2$, giving $k=14$. Option A is correct. Option B, −14, would make the sum 7/2 and −2 times the sum −7, not the product 7. Option C, 7, gives sum −7/4, so the condition fails. Option D, −7, gives sum 7/4, and again −2 times it is −7/2, not 7. Do not confuse the sum $-k/4$ with the product 7. Memory cue: for $ax^2+bx+c$, sum is $-b/a$ and product is $c/a$.

Q: The length of a rectangle is $x+3$ cm and its breadth is $x-2$ cm. If its area is $40$ cm², which is the correct quadratic equation in $x$?

$x^2+x-46=0$ For area, write $(x+3)(x-2)=40$. Expanding gives $x^2+x-6=40$, so $x^2+x-46=0$. Do not forget to move $-6$ to the other side.

Q: If $a=-2$, what type of statement does $(a+2)x^2+(a^2-4)x+7=0$ become?

Contradictory statement Substituting $a=-2$ gives $a+2=0$ and $a^2-4=4-4=0$. Therefore, the expression becomes $0x^2+0x+7=0$, or $7=0$. Since this statement is impossible, it is contradictory. Exam tip: when the coefficients of all variable terms become zero, classify the remaining constant statement rather than calling it linear or quadratic.

Question 1

What is the nature of the roots of the quadratic equation $16x^2-24kx+9k^2=0$?

Accepted Answer

Two equal real roots The equation can be written as $(4x-3k)^2=0$. Hence $4x-3k=0$, giving the repeated root $x=\frac{3k}{4}$. Equivalently, the discriminant is $D=b^2-4ac=0$, which confirms that the roots are equal and real. Exam tip: for a quadratic equation, $D=0$ indicates two equal real roots.

Question 2

If one root of (x^2+mx+36=0) is the reciprocal of the other, what can be said about (m)?

Accepted Answer

It is not possible If one root is the reciprocal of the other, the product of roots must be (1). Here the product is (36), so it is not possible.

Question 3

The roots of (x^2+px+25=0) are equal and positive. What is the value of (p)?

Accepted Answer

(-10) For equal roots, (p^2-100=0) gives (p=\pm10). For equal positive roots, (-\frac{p}{2}>0), so (p=-10).

Question 4

If (x=2) is not a root of (kx^2-7x+3=0), what condition must (k) satisfy?

Accepted Answer

(k
eq \frac{11}{4}) Putting (x=2), the left side becomes (4k-14+3=4k-11). For it not to be a root, (4k-11
eq0), so (k
eq\frac{11}{4}).

Question 5

Which quadratic equation has sum of roots (0) and product (-121)?

Accepted Answer

(x^2-121=0) The monic equation is (x^2-(	ext{sum})x+	ext{product}=0). Using sum (0) and product (-121) gives (x^2-121=0).

Question 6

If in $4x^2+kx+28=0$, the product of roots is (-2) times the sum of roots, what is (k)?

Accepted Answer

(14) The direct answer is option A, $k=14$. For $4x^2+kx+28=0$, let the roots be $\alpha$ and $\beta$. By the relations between coefficients and roots, their sum is $\alpha+\beta=-k/4$, and their product is $\alpha\beta=28/4=7$. The question says the product is −2 times the sum, so $7=-2(-k/4)$. The right side simplifies to $k/2$, giving $k=14$. Option A is correct. Option B, −14, would make the sum 7/2 and −2 times the sum −7, not the product 7. Option C, 7, gives sum −7/4, so the condition fails. Option D, −7, gives sum 7/4, and again −2 times it is −7/2, not 7. Do not confuse the sum $-k/4$ with the product 7. Memory cue: for $ax^2+bx+c$, sum is $-b/a$ and product is $c/a$.

Question 7

What is the sum of cubes of the roots of (x^2-5x-14=0)?

Accepted Answer

(335) The sum of roots is (5) and the product is (-14). (\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=125+210=335).

Question 8

The length of a rectangle is $x+3$ cm and its breadth is $x-2$ cm. If its area is $40$ cm², which is the correct quadratic equation in $x$?

Accepted Answer

$x^2+x-46=0$ For area, write $(x+3)(x-2)=40$. Expanding gives $x^2+x-6=40$, so $x^2+x-46=0$. Do not forget to move $-6$ to the other side.

Question 9

If $a=-2$, what type of statement does $(a+2)x^2+(a^2-4)x+7=0$ become?

Accepted Answer

Contradictory statement Substituting $a=-2$ gives $a+2=0$ and $a^2-4=4-4=0$. Therefore, the expression becomes $0x^2+0x+7=0$, or $7=0$. Since this statement is impossible, it is contradictory. Exam tip: when the coefficients of all variable terms become zero, classify the remaining constant statement rather than calling it linear or quadratic.

Question 10

What is the standard form of ((x-3)^2+(x+4)^2=(x-1)^2)?

Accepted Answer

(x^2+4x+24=0) Expanding gives left side (2x^2+2x+25) and right side (x^2-2x+1). Subtracting gives (x^2+4x+24=0).

Question 11

If $x=-1$ is a root of the quadratic equation $x^2+(4k-1)x+3k=0$, what is the value of $k$?

Accepted Answer

2 Since $x=-1$ is a root, substitute it into the equation: $(-1)^2+(4k-1)(-1)+3k=0$. This gives $1-4k+1+3k=0$, or $2-k=0$, so $k=2$. Exam tip: When a root is given, substitute it directly into the polynomial to form an equation in the parameter.

Question 12

The roots of the quadratic equation $x^2-7x+3=0$ are $\alpha$ and $\beta$. What is the value of $\alpha+\beta+3\alpha\beta$?

Accepted Answer

16 By Vieta’s formulas, for $ax^2+bx+c=0$, the sum of the roots is $-\frac{b}{a}$ and their product is $\frac{c}{a}$. Here, $a=1,b=-7,c=3$, so $\alpha+\beta=7$ and $\alpha\beta=3$. Therefore, $\alpha+\beta+3\alpha\beta=7+3(3)=16$. Option 7 is only the sum of the roots, while option 3 is only their product. Exam tip: Use Vieta’s formulas directly instead of solving for the roots.

Question 13

If the roots of (x^2+px+q=0) are (4) and (p), what is the value of (p)?

Accepted Answer

(-2) The direct answer is option A: p = -2. The equation is x² + px + q = 0, and its roots are 4 and p. For x² + bx + c = 0, the sum of roots equals -b. Here b = p, so the sum must be -p. From the stated roots, the sum is 4 + p. Equate the two descriptions of the same sum: 4 + p = -p. Add p to both sides: 4 + 2p = 0. Hence 2p = -4 and p = -2. Option A is correct. Option B, p = 2, would give 4 + p = 6, whereas -p = -2, so the root-sum rule would fail. Option C, p = -4, gives a sum of 0 but the coefficient rule gives 4, so it fails. Option D, p = 4, gives a sum of 8 but the coefficient rule gives -4, so it also fails. The symbol p appears both as a coefficient and as one root, so it must be handled consistently; it is not safe to treat those two appearances as unrelated. Exam cue: write “sum of roots = 4 + p = -p” before solving.

Question 14

If $\alpha$ and $\beta$ are the roots of the equation $x^2-12x+35=0$, what is the value of $(\alpha-5)(\beta-5)$?

Accepted Answer

0 By Vieta’s formulas, $\alpha+\beta=12$ and $\alpha\beta=35$. Therefore, $(\alpha-5)(\beta-5)=\alpha\beta-5(\alpha+\beta)+25=35-5(12)+25=0$. Hence, option A is correct. Exam tip: For expressions involving the roots, first use the sum and product of roots; finding the roots individually is not necessary.

Question 15

A right triangle has a base of $(x+3)$ units, a height of $(x+7)$ units, and an area of 55 square units. Which quadratic equation represents this situation?

Accepted Answer

$x^2+10x-89=0$ The area of a triangle is $\frac{1}{2}\times\text{base}\times\text{height}$. Hence, $\frac{1}{2}(x+3)(x+7)=55$, so $(x+3)(x+7)=110$. Expanding gives $x^2+10x+21=110$, and therefore $x^2+10x-89=0$. Option B results from mishandling the factor $\frac{1}{2}$ in the area formula. Exam tip: In triangle-area problems, first multiply both sides by 2 and then expand the product.

Question 16

If $\alpha$ and $\beta$ are the roots of $x^2-13x+36=0$, what is the value of $(\alpha+\beta)^2$?

Accepted Answer

169 For a quadratic equation $ax^2+bx+c=0$, the sum of the roots is $\alpha+\beta=-\frac{b}{a}$. Here, $a=1$ and $b=-13$, so $\alpha+\beta=13$. Therefore, $(\alpha+\beta)^2=13^2=169$, making option A correct. Option B is the constant term, not the sum of the roots. Exam tip: Use Vieta’s formulas directly: the sum of roots is $-\frac{b}{a}$ and their product is $\frac{c}{a}$.

Question 17

The equation $x^2+10x+k=0$ has no real roots. What is the correct condition on $k$?

Accepted Answer

$k>25$ A quadratic equation has no real roots when its discriminant satisfies $D<0$. Here, $D=b^2-4ac=10^2-4(1)(k)=100-4k$. Thus, $100-4k<0$, which gives $k>25$. When $k=25$, the discriminant is zero and the equation has one repeated real root, so that option is not correct. Exam tip: For “no real roots,” apply the condition $D<0$.

Question 18

If $x^2-2ax+a^2-25=0$, what is the difference between the larger and smaller roots?

Accepted Answer

10 The equation can be rewritten as $(x-a)^2-25=0$. Thus, $(x-a)^2=25$, giving the roots $a+5$ and $a-5$. Therefore, the difference between the larger and smaller roots is $(a+5)-(a-5)=10$. In such questions, completing the square is the quickest method.

Question 19

What is the standard form with integer coefficients of (\frac{(x+1)^2}{2}+\frac{(x-3)^2}{3}=10)?

Accepted Answer

(5x^2-6x-39=0) Multiplying the whole equation by (6) gives (3(x+1)^2+2(x-3)^2=60). Simplifying gives the correct form (5x^2-6x-39=0).

Question 20

If the roots of the equation $x^2-2(k+1)x+k^2=0$ are real and distinct, what is the correct condition on $k$?

Accepted Answer

$k>-rac{1}{2}$ A quadratic equation has two real and distinct roots only when its discriminant satisfies $D>0$. Here, $a=1$, $b=-2(k+1)$, and $c=k^2$, so $D=b^2-4ac=4(k+1)^2-4k^2=4(2k+1)$. Therefore, $4(2k+1)>0$, giving $k>-rac{1}{2}$. At $k=-\frac{1}{2}$, the roots are equal, so option B is not sufficient. Exam tip: use $D>0$ for distinct real roots, $D=0$ for equal roots, and $D<0$ for non-real roots.

Question 21

If (\alpha,\beta) are roots of (x^2-10x+q=0) and (\alpha^2+\beta^2=58), what is the value of (q)?

Accepted Answer

(21) Here (\alpha+\beta=10) and (\alpha\beta=q). Using (\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta), (58=100-2q), so (q=21).

Question 22

The length of a rectangle is $2x+1$ units and its breadth is $x-4$ units. If its area is $45$ square units, which quadratic equation represents this situation?

Accepted Answer

$2x^2-7x-49=0$ The area of a rectangle is length × breadth, so $(2x+1)(x-4)=45$. Expanding the product gives $2x^2-8x+x-4=2x^2-7x-4$. Therefore, $2x^2-7x-4=45$, which simplifies to $2x^2-7x-49=0$. Hence, option A is correct. Exam tip: For area-based questions, first form the product equation and then bring all terms to one side to make it equal to zero.

Question 23

What is the standard quadratic form of ((3x-2)(2x+5)=4(x+1))?

Accepted Answer

(6x^2+7x-14=0) The direct answer is A: $6x^2+7x-14=0$. First expand the left side: $(3x-2)(2x+5)=6x^2+15x-4x-10=6x^2+11x-10$. The right side is $4(x+1)=4x+4$. Move every term to the left: $6x^2+11x-10-4x-4=0$. Combining like terms gives $6x^2+7x-14=0$. Option A is correct. Option B forgets to subtract the right-side $4x$, so its linear coefficient is too large. Option C has the wrong sign for the constant term. Option D has both an incorrect linear coefficient and constant. Always expand first, then bring all terms to one side.

Question 24

If ((r^2-49)x^2+(r+7)x+2=0) is a quadratic equation, what is the correct condition on (r)?

Accepted Answer

(r
eq \pm7) For the equation to be quadratic, the coefficient of (x^2) must not be (0). Here (r^2-49
eq0), so (r
eq\pm7).

Question 25

What is the standard form of the equation 2(x−1)² + 3(x+2)² = 65?

Accepted Answer

5x² + 8x − 51 = 0 Expanding the brackets gives 2x² − 4x + 2 + 3x² + 12x + 12 = 65. Combining like terms, 5x² + 8x + 14 = 65, and moving 65 to the left gives 5x² + 8x − 51 = 0. Therefore, option A is correct. Option B has an incorrect sign for the linear term. Exam tip: Expand each squared binomial carefully before rearranging the equation into ax² + bx + c = 0.

Introduction to Quadratic Equations

Practice questions

01 What is the nature of the roots of the quadratic equation \(16x^2-24kx+9k^2=0\)?

02 If one root of (x^2+mx+36=0) is the reciprocal of the other, what can be said about (m)?

03 The roots of (x^2+px+25=0) are equal and positive. What is the value of (p)?

04 If (x=2) is not a root of (kx^2-7x+3=0), what condition must (k) satisfy?

05 Which quadratic equation has sum of roots (0) and product (-121)?

06 If in \(4x^2+kx+28=0\), the product of roots is (-2) times the sum of roots, what is (k)?

07 What is the sum of cubes of the roots of (x^2-5x-14=0)?

08 The length of a rectangle is \(x+3\) cm and its breadth is \(x-2\) cm. If its area is \(40\) cm², which is the correct quadratic equation in \(x\)?

09 If \(a=-2\), what type of statement does \((a+2)x^2+(a^2-4)x+7=0\) become?

10 What is the standard form of ((x-3)^2+(x+4)^2=(x-1)^2)?

11 If \(x=-1\) is a root of the quadratic equation \(x^2+(4k-1)x+3k=0\), what is the value of \(k\)?

12 The roots of the quadratic equation \(x^2-7x+3=0\) are \(\alpha\) and \(\beta\). What is the value of \(\alpha+\beta+3\alpha\beta\)?

13 If the roots of (x^2+px+q=0) are (4) and (p), what is the value of (p)?

14 If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2-12x+35=0\), what is the value of \((\alpha-5)(\beta-5)\)?

15 A right triangle has a base of \((x+3)\) units, a height of \((x+7)\) units, and an area of 55 square units. Which quadratic equation represents this situation?

16 If \(\alpha\) and \(\beta\) are the roots of \(x^2-13x+36=0\), what is the value of \((\alpha+\beta)^2\)?

17 The equation \\(x^2+10x+k=0\\) has no real roots. What is the correct condition on \\(k\\)?

18 If \(x^2-2ax+a^2-25=0\), what is the difference between the larger and smaller roots?

19 What is the standard form with integer coefficients of (\frac{(x+1)^2}{2}+\frac{(x-3)^2}{3}=10)?

20 If the roots of the equation \(x^2-2(k+1)x+k^2=0\) are real and distinct, what is the correct condition on \(k\)?

21 If (\alpha,\beta) are roots of (x^2-10x+q=0) and (\alpha^2+\beta^2=58), what is the value of (q)?

22 The length of a rectangle is \\(2x+1\\) units and its breadth is \\(x-4\\) units. If its area is \\(45\\) square units, which quadratic equation represents this situation?

23 What is the standard quadratic form of ((3x-2)(2x+5)=4(x+1))?

24 If ((r^2-49)x^2+(r+7)x+2=0) is a quadratic equation, what is the correct condition on (r)?

25 What is the standard form of the equation 2(x−1)² + 3(x+2)² = 65?

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