Class 11 Mathematics : Questions based on quadratic equations: Q4 – Q6

Q4 Find $k$ so that the quadratic equation $x^2-(k+3)x+k=0$ has equal and real roots.

Solution:

Step1 – For a quadratic $a{x}^2+bx+c=0$ here $a=1,b=-(k+3),c=k$.

Step 2 — Equal (repeated) real roots occur when the discriminant $\mathrm{\Delta }=b^2-4ac$ equals zero. Compute:

$$\mathrm{\Delta }=(-(k+3){)}^2-4\cdot 1\cdot k=(k+3{)}^2-4k\mathrm{.}$$

$$\mathrm{\Delta }=k^2+6k+9-4k=k^2+2k+9.$$

Step 3 — Solve $\mathrm{\Delta }=0$. Set $k^2+2k+9=0$. The discriminant of this quadratic in $k$ is $2^2-4\cdot 1\cdot 9=4-36=-32<0$, so there are no real solutions for $k$.

Conclusion: There is no real value of $k$ that makes the original quadratic have equal real roots. (If complex $k$ were allowed, $k=-1\pm 2i\sqrt{2}$.)

---

Q.5 Form a quadratic equation with integral coefficients whose roots are $2+\sqrt{5}$ and $2-\sqrt{5}$.

Solution:

Step 1 — If the roots are $r_1=2+\sqrt{5}$ and $r_2=2-\sqrt{5}$, then

$$\text{sum }=r_1+r_2=(2+\sqrt{5})+(2-\sqrt{5})=4,$$

$$\text{product }=r_1{r}_2=(2+\sqrt{5})(2-\sqrt{5})=4-5=-1.$$

Step 2 — A monic quadratic equation with these roots is

$$x^2-(\text{sum})x+(\text{product})=x^2-4x-1.$$Answer: quadratic equation is $\boxed{{\displaystyle x^2-4x-1=0}}$. Its coefficients are integer.

---

Q.6 For roots $\alpha ,\beta$ of $3x^2-7x+2=0$, compute ${\alpha }^2+{\beta }^2$.

Solution:

Step 1 — For $a{x}^2+bx+c=0$ we have sum= $\alpha +\beta =-{\displaystyle \frac{b}{a}}$ and product = $\alpha \beta ={\displaystyle \frac{c}{a}}$. Here $a=3,b=-7,c=2$, so

$$\alpha +\beta =-\frac{-7}{3}=\frac{7}{3},\alpha \beta =\frac{2}{3}\mathrm{.}$$

Step 2 — By identity.

$${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta \mathrm{.}$$

Substitute the values:

$${\alpha }^2+{\beta }^2={\left(\frac{7}{3}\right)}^2-2\cdot \frac{2}{3}=\frac{49}{9}-\frac{4}{3}\mathrm{.}$$

convert to a common denominator:

$$\frac{49}{9}-\frac{12}{9}=\frac{37}{9}\mathrm{.}$$

Answer : $\boxed{{\displaystyle {\alpha }^2+{\beta }^2=\frac{37}{9}}}$.