Q: If ((k^2-25)x^2+(k-5)x+1=0) is a quadratic equation, what is the correct condition on (k)?

(k\neq \pm5) For the equation to be quadratic, (k^2-25\neq0) is required. So both (k\neq5) and (k\neq-5) are necessary.

Q: What is the standard form $ax^2+bx+c=0$ of the equation $(x+2)^2+2(x-3)^2=35$?

$3x^2-8x-13=0$ Expand the squares: $(x+2)^2=x^2+4x+4$ and $2(x-3)^2=2x^2-12x+18$. Thus, $3x^2-8x+22=35$, and moving 35 to the left gives $3x^2-8x-13=0$. Therefore, option A is correct. Exam tip: when converting to standard form, keep all terms on one side and make the other side zero; the constant becomes $22-35=-13$.

Q: If $x=-3$ is a root of the equation $4x^2+px-9=0$, what is the value of $p$?

9 A root must satisfy the equation. Substituting $x=-3$ gives $4(-3)^2+p(-3)-9=0$, so $36-3p-9=0$. Hence, $27-3p=0$ and $p=9$. Therefore, option A is correct. Exam tip: Substitute the given root directly into the equation and solve for the unknown parameter.

Q: Which monic quadratic equation has sum of roots (-11) and product (30)?

(x^2+11x+30=0) A monic equation is (x^2-(\text{sum})x+\text{product}=0). Substituting sum (-11) gives (x^2+11x+30=0).

Q: If the roots of the quadratic equation $x^2-2mx+36=0$ are equal, what are the possible values of $m$?

$m=\pm6$ For equal roots, the discriminant must be zero. Here, $a=1$, $b=-2m$, and $c=36$, so $D=b^2-4ac=(-2m)^2-4(1)(36)=4m^2-144$. Setting $D=0$ gives $4m^2-144=0$, hence $m^2=36$ and $m=\pm6$. Choosing only $m=6$ or only $m=-6$ omits one valid possibility. Exam tip: whenever the roots are equal, set the discriminant equal to zero.

Q: If $p$ is a real number, what is the condition for the equation $x^2+2px+25=0$ to have real roots?

$p\leq -5$ या $p\geq 5$ A quadratic equation has real roots when its discriminant satisfies $D\geq 0$. Here, $a=1$, $b=2p$, and $c=25$, so $D=(2p)^2-4(1)(25)=4p^2-100$. Thus, $4p^2-100\geq 0$, which gives $p^2\geq 25$, and hence $p\leq -5$ or $p\geq 5$. Option B reverses the required inequality and also excludes the double real roots occurring at $p=\pm5$. Exam tip: For real roots, use $D\geq0$, not only $D>0$.

Q: If 5 and 6 are the roots of the equation $x^2-sx+p=0$, what is the value of $s+p$?

41 For the quadratic equation $x^2-sx+p=0$, the sum of the roots is $s$, and their product is $p$. Thus, $s=5+6=11$ and $p=5\times6=30$, so $s+p=11+30=41$. Exam tip: Compare the equation with $x^2-(\text{sum of roots})x+(\text{product of roots})$ to identify the parameters quickly.

Question 1

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

Accepted Answer

(6x^2+7x-16=0) Here ((2x+5)(3x-4)=6x^2+7x-20) and (4(x-1)=4x-4). Bringing all terms to one side gives (6x^2+3x-16=0).

Question 2

If ((k^2-25)x^2+(k-5)x+1=0) is a quadratic equation, what is the correct condition on (k)?

Accepted Answer

(k
eq \pm5) For the equation to be quadratic, (k^2-25
eq0) is required. So both (k
eq5) and (k
eq-5) are necessary.

Question 3

What is the standard form $ax^2+bx+c=0$ of the equation $(x+2)^2+2(x-3)^2=35$?

Accepted Answer

$3x^2-8x-13=0$ Expand the squares: $(x+2)^2=x^2+4x+4$ and $2(x-3)^2=2x^2-12x+18$. Thus, $3x^2-8x+22=35$, and moving 35 to the left gives $3x^2-8x-13=0$. Therefore, option A is correct. Exam tip: when converting to standard form, keep all terms on one side and make the other side zero; the constant becomes $22-35=-13$.

Question 4

What is the standard form with integer coefficients of (\frac{2x^2-1}{5}+\frac{x+3}{2}=7)?

Accepted Answer

(4x^2+5x-59=0) Multiplying the whole equation by (10) gives (4x^2-2+5x+15=70). Therefore the standard form is (4x^2+5x-57=0).

Question 5

If $x=-3$ is a root of the equation $4x^2+px-9=0$, what is the value of $p$?

Accepted Answer

9 A root must satisfy the equation. Substituting $x=-3$ gives $4(-3)^2+p(-3)-9=0$, so $36-3p-9=0$. Hence, $27-3p=0$ and $p=9$. Therefore, option A is correct. Exam tip: Substitute the given root directly into the equation and solve for the unknown parameter.

Question 6

Which monic quadratic equation has sum of roots (-11) and product (30)?

Accepted Answer

(x^2+11x+30=0) A monic equation is (x^2-(	ext{sum})x+	ext{product}=0). Substituting sum (-11) gives (x^2+11x+30=0).

Question 7

If the roots of the quadratic equation $x^2-2mx+36=0$ are equal, what are the possible values of $m$?

Accepted Answer

$m=\pm6$ For equal roots, the discriminant must be zero. Here, $a=1$, $b=-2m$, and $c=36$, so $D=b^2-4ac=(-2m)^2-4(1)(36)=4m^2-144$. Setting $D=0$ gives $4m^2-144=0$, hence $m^2=36$ and $m=\pm6$. Choosing only $m=6$ or only $m=-6$ omits one valid possibility. Exam tip: whenever the roots are equal, set the discriminant equal to zero.

Question 8

If $p$ is a real number, what is the condition for the equation $x^2+2px+25=0$ to have real roots?

Accepted Answer

$p\leq -5$ या $p\geq 5$ A quadratic equation has real roots when its discriminant satisfies $D\geq 0$. Here, $a=1$, $b=2p$, and $c=25$, so $D=(2p)^2-4(1)(25)=4p^2-100$. Thus, $4p^2-100\geq 0$, which gives $p^2\geq 25$, and hence $p\leq -5$ or $p\geq 5$. Option B reverses the required inequality and also excludes the double real roots occurring at $p=\pm5$. Exam tip: For real roots, use $D\geq0$, not only $D>0$.

Question 9

What is the sum of reciprocals of the roots of (4x^2-13x+9=0)?

Accepted Answer

( \frac{13}{9} ) The sum of reciprocals is (\frac{\alpha+\beta}{\alpha\beta}). Here it is (\frac{\frac{13}{4}}{\frac{9}{4}}=\frac{13}{9}).

Question 10

If 5 and 6 are the roots of the equation $x^2-sx+p=0$, what is the value of $s+p$?

Accepted Answer

41 For the quadratic equation $x^2-sx+p=0$, the sum of the roots is $s$, and their product is $p$. Thus, $s=5+6=11$ and $p=5\times6=30$, so $s+p=11+30=41$. Exam tip: Compare the equation with $x^2-(\text{sum of roots})x+(\text{product of roots})$ to identify the parameters quickly.

Question 11

If the roots of the equation $x^2-12x+k=0$ are real and distinct, what is the correct condition on $k$?

Accepted Answer

$k<36$ For a quadratic equation $ax^2+bx+c=0$, the roots are real and distinct only when the discriminant $D=b^2-4ac$ is greater than zero. Here, $a=1$, $b=-12$, and $c=k$, so $D=(-12)^2-4(1)(k)=144-4k>0$. Therefore, $k<36$. When $k=36$, the roots are equal, so that option is not correct. Exam tip: For “real and distinct” roots, always apply $D>0$.

Question 12

Which of the following quadratic equations has roots that are reciprocals of each other?

Accepted Answer

$x^2-5x+1=0$ If roots $\alpha,\beta$ are reciprocals, then $\alpha\beta=1$. For $ax^2+bx+c=0$, their product is $c/a$. In option A, $c/a=1/1=1$; in B it is $1/2$. Exam tip: check $c/a$ directly.

Question 13

What is the sum of squares of the roots of $5x^2+6x+1=0$?

Accepted Answer

$ \frac{26}{25} $ $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$. Here $\left(-\frac{6}{5}\right)^2-2\cdot\frac{1}{5}=\frac{26}{25}$.

Question 14

If α and β are the roots of the equation x² − 9x + 18 = 0, what is the value of (α − 3)(β − 3)?

Accepted Answer

0 By Vieta’s formulas, α + β = 9 and αβ = 18. Thus, (α − 3)(β − 3) = αβ − 3(α + β) + 9 = 18 − 27 + 9 = 0. Therefore, option A is correct.

Question 15

Which of the following is a quadratic equation, even though it has terms involving the variable on both sides?

Accepted Answer

x^2+4x=3x+7 In option B, bringing all terms to one side gives $x^2+4x-3x-7=0$, or $x^2+x-7=0$. Its highest power is 2, so it is quadratic. Option A is linear. Exam tip: first rewrite the equation as $ax^2+bx+c=0$.

Question 16

Which statement correctly describes the nature of the roots of $x^2-8x+17=0$?

Accepted Answer

It has no real roots Here, $a=1$, $b=-8$, and $c=17$. Therefore, the discriminant is $\Delta=b^2-4ac=(-8)^2-4(1)(17)=64-68=-4$. Since $\Delta<0$, the equation has no real roots, so option A is correct. Option B would be correct only if $\Delta=0$. Exam tip: $\Delta<0$ means no real roots, $\Delta=0$ means equal real roots, and $\Delta>0$ means distinct real roots.

Question 17

If one root of the equation $x^2-(m+5)x+5m=0$ is $5$, what is the other root?

Accepted Answer

m For a quadratic equation, the product of the roots equals the constant term divided by the coefficient of $x^2$. Here, the product of the roots is $5m$. Since one root is $5$, the other root is $\frac{5m}{5}=m$. Exam tip: divide the product of the roots by the known root to obtain the other root.

Question 18

What is the absolute difference between the roots of $3x^2-14x+16=0$?

Accepted Answer

$\frac{2}{3}$ Factoring gives $(3x-8)(x-2)=0$, so the roots are $x=\frac{8}{3}$ and $x=2$. Therefore, their absolute difference is $\left|\frac{8}{3}-2\right|=\frac{2}{3}$. Exam tip: when the difference between roots is asked, use the larger root minus the smaller root unless an order is explicitly specified.

Question 19

Which condition is essential for an equation of the form $ax^2+bx+c=0$ to be called a quadratic equation?

Accepted Answer

$a\ne0$ The coefficient of $x^2$ must be non-zero, so $a\ne0$ is essential. If $a=0$, the equation becomes linear at most. Exam tip: identify the equation type by checking the highest power of the variable.

Question 20

If both roots of the quadratic equation $x^2+bx+64=0$ are equal and each root is $-8$, what is the value of $b$?

Accepted Answer

16 The roots are $-8$ and $-8$, so their sum is $-16$. For $x^2+bx+64=0$, the sum of the roots is $-b$. Therefore, $-b=-16$, giving $b=16$. Option B is the root sum, not the coefficient $b$. Exam tip: In $x^2+px+q=0$, the sum of roots is $-p$ and their product is $q$.

Question 21

In which option will the sum of roots be positive and the product of roots be negative?

Accepted Answer

(x^2-6x-16=0) In the first option, the sum is (-\frac{b}{a}=6) and the product is (\frac{c}{a}=-16). So the sum is positive and the product is negative.

Question 22

A rectangle has length $(x+8)$ and breadth $(x-5)$. If its area is $104$ square units, which quadratic equation correctly represents this situation?

Accepted Answer

$x^2+3x-144=0$ Area of a rectangle = length × breadth, so $(x+8)(x-5)=104$. On expanding, we get $x^2+3x-40=104$. Bringing all terms to one side gives $x^2+3x-144=0$, so option A is correct. Option B results from failing to transfer the $-40$ term correctly while simplifying. Exam tip: For area-based word problems, first write length × breadth = given area, then rearrange the equation into standard quadratic form.

Question 23

Two numbers have sum (15) and product (54). They can be roots of which quadratic equation?

Accepted Answer

(x^2-15x+54=0) If the sum of roots is (15) and product is (54), the equation is (x^2-15x+54=0). Remember the monic form formula.

Question 24

If (\alpha,\beta) are roots of (x^2-11x+30=0), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

Accepted Answer

( \frac{11}{30} ) (\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}). Here the value is (\frac{11}{30}).

Question 25

If $x+a$^2=x^2+18x+81 is an identity, what is the value of $a$?

Accepted Answer

9 Expanding gives $x+a$^2=x^2+2ax+a^2"). Comparing the coefficients of $x$, we get $2a=18$, so $a=9$. This also satisfies the constant-term condition $a^2=81$. Although $-9$ gives the same constant term, it makes the coefficient of $x$ equal to $-18$, so it is incorrect. Exam tip: For an identity, compare coefficients of like powers on both sides.

Introduction to Quadratic Equations

Practice questions

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

02 If ((k^2-25)x^2+(k-5)x+1=0) is a quadratic equation, what is the correct condition on (k)?

03 What is the standard form \(ax^2+bx+c=0\) of the equation \((x+2)^2+2(x-3)^2=35\)?

04 What is the standard form with integer coefficients of (\frac{2x^2-1}{5}+\frac{x+3}{2}=7)?

05 If \(x=-3\) is a root of the equation \(4x^2+px-9=0\), what is the value of \(p\)?

06 Which monic quadratic equation has sum of roots (-11) and product (30)?

07 If the roots of the quadratic equation \(x^2-2mx+36=0\) are equal, what are the possible values of \(m\)?

08 If \(p\) is a real number, what is the condition for the equation \(x^2+2px+25=0\) to have real roots?

09 What is the sum of reciprocals of the roots of (4x^2-13x+9=0)?

10 If 5 and 6 are the roots of the equation \(x^2-sx+p=0\), what is the value of \(s+p\)?

11 If the roots of the equation \(x^2-12x+k=0\) are real and distinct, what is the correct condition on \(k\)?

12 Which of the following quadratic equations has roots that are reciprocals of each other?

13 What is the sum of squares of the roots of \(5x^2+6x+1=0\)?

14 If α and β are the roots of the equation x² − 9x + 18 = 0, what is the value of (α − 3)(β − 3)?

15 Which of the following is a quadratic equation, even though it has terms involving the variable on both sides?

16 Which statement correctly describes the nature of the roots of \(x^2-8x+17=0\)?

17 If one root of the equation \(x^2-(m+5)x+5m=0\) is \(5\), what is the other root?

18 What is the absolute difference between the roots of \(3x^2-14x+16=0\)?

19 Which condition is essential for an equation of the form \(ax^2+bx+c=0\) to be called a quadratic equation?

20 If both roots of the quadratic equation \\(x^2+bx+64=0\\) are equal and each root is \\(-8\\), what is the value of \\(b\\)?

21 In which option will the sum of roots be positive and the product of roots be negative?

22 A rectangle has length \((x+8)\) and breadth \((x-5)\). If its area is \(104\) square units, which quadratic equation correctly represents this situation?

23 Two numbers have sum (15) and product (54). They can be roots of which quadratic equation?

24 If (\alpha,\beta) are roots of (x^2-11x+30=0), what is (\frac{1}{\alpha}+\frac{1}{\beta})?

25 If \\(x+a\\)^2=x^2+18x+81 is an identity, what is the value of \\(a\\)?

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