Q: What is the coefficient of $x$ in the equation $2x^2 - x + 8 = 0$?

-1 In the equation $2x^2 - x + 8 = 0$ the linear term is $-x$. The coefficient is the number multiplied by $x$, so here it is $-1$. Hence the correct answer is $-1$. Option B (1) is incorrect because it ignores the negative sign; options C and D are unrelated values. Exam tip: First identify the linear term and always include its sign when stating the coefficient.

Q: What is the standard form of $x(x-5)=0$?

$x^2-5x=0$ Expanding gives $x(x-5)=x\cdot x - x\cdot 5 = x^2-5x$, so the standard form is $x^2-5x=0$. Option A has the wrong sign (should be -5x, not +5x); option C has incorrect coefficients; option D incorrectly changes the linear term to a constant. Exam tip: apply distributive property term-by-term and pay attention to signs when removing brackets.

Q: What is the standard form of the equation $x^2 = 6x$?

$x^2-6x=0$ Bring the term on the right, $6x$, to the left; the sign changes giving $x^2-6x=0$. The standard form is $ax^2+bx+c=0$ with all terms on one side. Option A has the wrong sign for the $x$-term; options C and D do not represent the same rearrangement of the original equation. Exam tip: always move all terms to one side so the equation equals zero and then identify $a, b, c$.

Question 1

Is $x=1$ a root of the equation $x^2 - x = 0$?

Accepted Answer

Yes To check a root substitute the value into the equation: $1^2-1=1-1=0$. Thus $x=1$ satisfies the equation and is a root. The closest distractor "Only $x=0$ is a root" is incorrect because factoring gives $x(x-1)=0$, so both $x=0$ and $x=1$ are roots. Exam tip: Either substitute the candidate value or factor the quadratic to identify roots quickly.

Question 2

What is the expression (p(x)=x^2+2x+1)?

Accepted Answer

Quadratic polynomial (p(x)=x^2+2x+1) is a polynomial because it is not set equal to (0). It becomes an equation when written as equal to (0).

Question 3

What is the degree of the equation (x^2+1=0)?

Accepted Answer

2 The degree of a polynomial is the highest exponent of the variable with a nonzero coefficient. The left-hand side polynomial x^2+1 has highest power 2, so the degree is 2. Option A (1) is incorrect because that would require a highest power of x to be 1; option C (0) would apply only to a nonzero constant polynomial; option D (4) is wrong since there is no x^4 term. Exam tip: To find the degree of an equation, look at the polynomial on the left side in standard form and ignore constant terms—take the largest exponent of the variable.

Question 4

What is the coefficient of $x$ in the equation $2x^2 - x + 8 = 0$?

Accepted Answer

-1 In the equation $2x^2 - x + 8 = 0$ the linear term is $-x$. The coefficient is the number multiplied by $x$, so here it is $-1$. Hence the correct answer is $-1$. Option B (1) is incorrect because it ignores the negative sign; options C and D are unrelated values. Exam tip: First identify the linear term and always include its sign when stating the coefficient.

Question 5

What is the value of (c) in (x^2+7x=0)?

Accepted Answer

(0) The correct answer is option D, c=0. A quadratic equation is commonly written as ax^2+bx+c=0, where c is the constant term, the term without x. In x^2+7x=0, the coefficient of x^2 is 1, the coefficient of x is 7, and there is no separate constant term. A missing term has coefficient zero, so the equation can be written as x^2+7x+0=0. Therefore c=0. Option A, 1, is the coefficient of x^2, so it is a, not c. Option B, 7, is the coefficient of x, so it is b, not c. Option C, −7, is not present as any coefficient in the given equation. Option D correctly represents the absent constant term. A helpful check is to compare each term with its position in ax^2+bx+c=0: x^2 gives a, x gives b, and the number alone gives c.

Question 6

What is the standard form of $x(x-5)=0$?

Accepted Answer

$x^2-5x=0$ Expanding gives $x(x-5)=x\cdot x - x\cdot 5 = x^2-5x$, so the standard form is $x^2-5x=0$. Option A has the wrong sign (should be -5x, not +5x); option C has incorrect coefficients; option D incorrectly changes the linear term to a constant. Exam tip: apply distributive property term-by-term and pay attention to signs when removing brackets.

Question 7

What is the standard form of the equation $x^2 = 6x$?

Accepted Answer

$x^2-6x=0$ Bring the term on the right, $6x$, to the left; the sign changes giving $x^2-6x=0$. The standard form is $ax^2+bx+c=0$ with all terms on one side. Option A has the wrong sign for the $x$-term; options C and D do not represent the same rearrangement of the original equation. Exam tip: always move all terms to one side so the equation equals zero and then identify $a, b, c$.

Question 8

Which equation has the highest power, 2, of the variable?

Accepted Answer

3x² + 1 = 0 The degree of an equation in one variable is determined by the greatest exponent of that variable. In option A, the degree is 1; in option B, it is 3; in option C, the greatest exponent is 2; and option D contains a square root, equivalent to x^(1/2). Therefore, 3x² + 1 = 0 has highest power 2 and is a quadratic equation. Hence option C is correct.

Question 9

Why is (\frac{1}{x^2}+x+1=0) not a quadratic equation?

Accepted Answer

Because it has a negative power of (x) (\frac{1}{x^2}=x^{-2}), which is not polynomial form. A quadratic equation cannot have a negative power of the variable.

Question 10

Why is (\sqrt{x}+x^2=0) not considered a quadratic equation in the usual form?

Accepted Answer

Because it has a (\sqrt{x}) term The term (\sqrt{x}) shows a fractional power of the variable, so it is not in usual quadratic form. A quadratic equation has only (x^2), (x), and constant terms.

Question 11

What is the standard form of $(x+1)^2 = 0$?

Accepted Answer

$x^2 + 2x + 1 = 0$ Expanding $(x+1)^2$ gives $x^2 + 2x + 1$. The standard quadratic form places all terms on one side equal to zero, so the correct form is $x^2 + 2x + 1 = 0$. Option B is incorrect because $x^2 - 2x + 1$ is the expansion of $(x-1)^2$; options A and D omit or alter the linear term. Exam tip: use the identity $(a+b)^2 = a^2 + 2ab + b^2$ to expand quickly and then set the expression equal to zero.

Question 12

If $x=2$ is a root of the quadratic equation $x^2+kx+2=0$, what is the value of $k$?

Accepted Answer

-3 Substitute the root $x=2$ into the polynomial: $2^2+k\cdot2+2=0$ so $4+2k+2=0$. Thus $2k=-6$ and $k=-3$. Option A (3) is incorrect because substituting 3 gives $4+6+2=12\neq0$. Option D ($-2$) is also wrong since it yields $4-4+2=2\neq0$. Exam tip: use direct substitution or the Factor Theorem — if $x=a$ is a root then the polynomial evaluated at $a$ equals zero.

Question 13

If $a=3,\; b=0,\; c=-12$, which is the corresponding quadratic equation?

Accepted Answer

$3x^2-12=0$ Use the standard form $ax^2+bx+c=0$. Substituting gives $3x^2+0x-12=0$, and since the $x$-term is absent this simplifies to $3x^2-12=0$. You can further divide by 3 to get $x^2-4=0$. Closest distractors fail for clear reasons: option C wrongly includes a $-12x$ term (but $b=0$), option B has the wrong sign on the constant, and option D swaps coefficients. Exam tip: always substitute coefficients carefully and then simplify; check whether the $x$-term should be present when $b=0$.

Question 14

Which term is absent in (2x^2+5=0)?

Accepted Answer

Linear term There is no (x) term, so the linear term is absent. The coefficient of a missing term is considered (0).

Question 15

Which of the following is a quadratic equation in (y)?

Accepted Answer

(2y^2-3y+1=0) The highest power of (y) is (2), so it is quadratic in (y). The variable may be different, but the degree must be (2).

Question 16

In the standard form $ax^2+bx+c=0$, which symbol denotes the coefficient of $x^2$?

Accepted Answer

a In the standard form $ax^2+bx+c=0$, the coefficient of $x^2$ is $a$. The coefficient is the number multiplying that power of $x$; if $a=0$ the equation would not be quadratic. Option A ($b$) is incorrect because $b$ is the coefficient of $x$; option C ($c$) is the constant term; option D ($x$) is the variable itself. Exam tip: always check that the leading coefficient $a$ is nonzero when identifying a quadratic equation.

Question 17

Is (x^2+4=0) a quadratic equation?

Accepted Answer

Yes because the coefficient of (x^2) is (1) In (x^2+4=0), the coefficient of (x^2) is (1), so it is quadratic. Having real roots is not a condition for being quadratic.

Question 18

In the standard quadratic equation (ax^2+bx+c=0), what is the maximum number of different terms?

Accepted Answer

(3) A quadratic equation has highest power 2, so its standard form is written as $ax^2+bx+c=0$, where $a\ne0$. This form can contain three kinds of terms: the quadratic term $ax^2$, the linear term $bx$, and the constant term $c$. If a coefficient is zero, one of these terms may disappear in a particular equation, but the maximum possible number of different terms remains three.

For example, $2x^2+5x+7=0$ contains all three terms. The symbols $a$, $b$, and $c$ are coefficients or constants attached to these terms, not extra terms themselves. Hence the maximum is 3, which is option C. Option 2 would apply only when one term is absent, while option 4 incorrectly counts something that is not a separate algebraic term.

Question 19

Is (2x^2=0) a quadratic equation?

Accepted Answer

Yes because the coefficient of (x^2) is (2) In (2x^2=0), the coefficient of (x^2) is (2
eq 0). It can be quadratic even without linear and constant terms.

Question 20

A rectangle has length (x+3) and breadth (x). If its area is (10), which quadratic equation is correct?

Accepted Answer

(x^2+3x-10=0) The area is (x(x+3)=10), which gives (x^2+3x-10=0). In geometry questions, form the equation using the formula.

Question 21

If a = −2, b = 9 and c = −7, which is the corresponding quadratic equation?

Accepted Answer

−2x² + 9x − 7 = 0 A quadratic equation in standard form is ax² + bx + c = 0. The values supplied are a = −2, b = 9 and c = −7. Substituting them in their respective positions gives (−2)x² + 9x + (−7) = 0, which simplifies to −2x² + 9x − 7 = 0. Therefore option B is correct. The coefficient a belongs to x², b belongs to x, and c is the constant term; changing their positions produces a different equation. Option A loses the negative sign of a, option C changes the sign of b, and option D swaps the roles of a and c. Keeping the sign and position of every coefficient is essential.

Introduction to Quadratic Equations

Practice questions

01 Is \(x=1\) a root of the equation \(x^2 - x = 0\)?

02 What is the expression (p(x)=x^2+2x+1)?

03 What is the degree of the equation (x^2+1=0)?

04 What is the coefficient of \(x\) in the equation \(2x^2 - x + 8 = 0\)?

05 What is the value of (c) in (x^2+7x=0)?

06 What is the standard form of \(x(x-5)=0\)?

07 What is the standard form of the equation \(x^2 = 6x\)?

08 Which equation has the highest power, 2, of the variable?

09 Why is (\frac{1}{x^2}+x+1=0) not a quadratic equation?

10 Why is (\sqrt{x}+x^2=0) not considered a quadratic equation in the usual form?

11 What is the standard form of \((x+1)^2 = 0\)?

12 If \(x=2\) is a root of the quadratic equation \(x^2+kx+2=0\), what is the value of \(k\)?

13 If \(a=3,\; b=0,\; c=-12\), which is the corresponding quadratic equation?

14 Which term is absent in (2x^2+5=0)?

15 Which of the following is a quadratic equation in (y)?

16 In the standard form \(ax^2+bx+c=0\), which symbol denotes the coefficient of \(x^2\)?

17 Is (x^2+4=0) a quadratic equation?

18 In the standard quadratic equation (ax^2+bx+c=0), what is the maximum number of different terms?

19 Is (2x^2=0) a quadratic equation?

20 A rectangle has length (x+3) and breadth (x). If its area is (10), which quadratic equation is correct?

21 If a = −2, b = 9 and c = −7, which is the corresponding quadratic equation?

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