# Class 10 Math Theory of Quadratic Equations 4. Find the value of k if the roots of the following equations are equal.(iii) (3 k+2) x^{2}-5(k+1) x+(2 k+3)=0

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##### 4. Find the value of k if the roots of the following equations are equal.(iii) (3 k+2) x^{2}-5(k+1) x+(2 k+3)=0

(x) \alpha^{2}+\beta^{2} is equal to(a) \alpha^{2}-\beta^{2} (b) \frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}} (c) (\alpha+\beta)^{2}-2 \alpha \beta (d) \alpha+\beta

(ix) \frac{1}{\alpha}+\frac{1}{\beta} is equal to(a) \frac{1}{\alpha} (b) \frac{1}{\alpha}-\frac{1}{\beta} (c) \frac{\alpha-\beta}{\alpha \beta} (d). \frac{\alpha+\beta}{\alpha \beta}

4. Find p if(ii) the roots of the equation x^{2}+3 x+p-2=0 differ by 2 .

2 . Evaluate(i) \left(1-\omega-\omega^{2}\right)^{7}

1. Without solving find the sum and the product of the roots of the following ratic equations.(ii) 3 x^{2}+7 x-11=0

6. Find m if sum and product of the roots of the following equations is equal to a given number \lambda .(ii) 4 x^{2}-(3+5 m) x-(9 m-17)=0

(xii) Roots of the equation 4 x^{2}-4 x+1=0 are(a) real equal(b) real unequal (c) imaginary(d) irrational

Example 1: Solve the system of equations 3 x+y=4 and 3 x^{2}+y^{2}=52 .

2. Evaluate (iv) \left(2+2 \omega-2 \omega^{2}\right)\left(3-3 \omega+3 \omega^{2}\right)

4. Find the value of k if the roots of the following equations are equal.(ii) x^{2}+2(k+2) x+(3 k+4)=0

1. Find the discriminant of the following given ratic equations:(ii) 6 x^{2}-8 x+3=0

1. Find the discriminant of the following given ratic equations:(iv) 4 x^{2}-7 x-2=0

2. Write short answers of the following questions.(i) Discuss the nature of the roots of the following equations.(a) x^{2}+3 x+5=0

2. If \alpha \beta are the roots of the equation x^{2}-3 x+6=0 .Form equations whose roots are(c) \frac{1}{\alpha} \frac{1}{\beta}

Example I: Iff c_{k} [pare the rooss of the ratic equation

1. Multiple Choice QuestionsFour possible answers are given for the following questions. Tick (\checkmark) the correct answer.(i) If \alpha \beta are the roots of 3 x^{2}+5 x-2=0 then \alpha+\beta is(a) \frac{5}{3} (b) \frac{3}{5} (c) \frac{-5}{3} (d) \frac{-2}{3}

Examples 3: Find k if the roots of the equation (k+3) x^{2}-2(k+1) x-(k+1)=0 are equal if k \neq-3 .

1. Find the discriminant of the following given ratic equations:(iii) 9 x^{2}-30 x+25=0

4. Solve by using synthetic division if(ii) 3 is the root of the equation 2 x^{3}-3 x^{2}-11 x+6=0

2. Write short answers of the following questions.(vi) Evaluate \omega^{37}+\omega^{38}+1

1. Write the ratic equations having following roots.(b) 49

2. Find the nature of the roots of the following given ratic equations and verify the result by solving the equations:(i) x^{2}-23 x+120=0

Example 2: If \alpha \beta are the roots of the equation 2 x^{2}+3 x+4=0 thenfind the value of(ii) \frac{1}{\alpha}+\frac{1}{\beta}

2. Find the value of k if(i) sum of the roots of the equation 2 k x^{2}-3 x+4 k=0 is twice the product of the roots.

(xiii) If \alpha \beta are the roots of 4 x^{2}-3 x+6=0 find(a) \alpha^{2}+\beta^{2} (b) \frac{\alpha}{\beta}+\frac{\beta}{\alpha} (c) \alpha-\beta

(viii) If b^{2}-4 a c>0 but not a perfect square then roots of a x^{2}+b x+c=0 are(a) imaginary(b) rational(c) irrational(d) none of these

3. Use synthetic division to find the values of l and m if(ii) (x-1) and (x+1) are the factors of the polynomial x^{3}-3 l x^{2}+2 m x+6

Example 1: Find the value of h if the sum of the roots is equal to 3 -times the product of the roots of the equation 3 x^{2}+(9-6 h) x+5 h=0 .