If the roots of the equations $ax^2+2bx+c=0$ and $bx^2-2\sqrt{ac}x+b=0$ are simultaneously real, then prove that $b^2=ac$.
Given:
Given quadratic equations are $ax^2+2bx+c=0$ and $bx^2-2\sqrt{ac}x+b=0$. The roots of the given quadratic equations are simultaneously real.
To do:
We have to prove that $b^2=ac$.
Solution:
Let $D_1$ be the discriminant of $ax^2+2bx+c=0$ and $D_2$ be the discriminant of $bx^2-2\sqrt{ac}x+b=0$.
The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is $D=b^2-4ac$.
Therefore,
$D_1=(2b)^2-4(a)(c)$
$D_1=4b^2-4ac$
$D_2=(-2\sqrt{ac})^2-4(b)(b)$
$D_2=4ac-4b^2$
The given quadratic equations have real roots if $D_1≥0$ and $D_2≥0$.
This implies,
$4b^2-4ac≥0$ and $4ac-4b^2≥0$
$b^2-ac≥0$ and $ac-b^2≥0$
$b^2≥ac$ and $ac≥b^2$
Therefore,
$b^2=ac$
Hence proved.
Related Articles
- If $a, b, c$ are real numbers such that $ac≠0$, then show that at least one of the equations $ax^2+bx+c=0$ and $-ax^2+bx+c=0$ has real roots.
- If $\alpha,\ \beta$ are the roots of the equation $ax^2+bx+c=0$, then find the roots of the equation $ax^2+bx(x+1)+c(x+1)^2=0$.
- If the roots of the equation $(a^2+b^2)x^2-2(ac+bd)x+(c^2+d^2)=0$ are equal, prove that $\frac{a}{b}=\frac{c}{d}$.
- If $ad \neq bc$, then prove that the equation $\left( a^{2} +b^{2}\right) x^{2} +2( ac\ +\ bd) \ x+\left( c^{2} +d^{2}\right) =0$ has no real roots.
- If the roots of the equation $(c^2-ab)x^2-2(a^2-bc)x+b^2-ac=0$ are equal, prove that either $a=0$ or $a^3+b^3+c^3=3abc$.
- If \( x=\frac{\sqrt{a^{2}+b^{2}}+\sqrt{a^{2}-b^{2}}}{\sqrt{a^{2}+b^{2}}-\sqrt{a^{2}-b^{2}}} \), then prove that \( b^{2} x^{2}-2 a^{2} x+b^{2}=0 \).
- If $ax+by=a^{2}-b^{2}$ and $bx+ay=0$, then find the value of $( x+y)$.
- If the angles $A,\ B,\ C$ of a $\vartriangle ABC$ are in A.P., then prove that $b^2=a^2+c^2-ac$.
- Choose the correct answer from the given four options in the following questions:Which of the following equations has no real roots?(A) \( x^{2}-4 x+3 \sqrt{2}=0 \)(B) \( x^{2}+4 x-3 \sqrt{2}=0 \)(C) \( x^{2}-4 x-3 \sqrt{2}=0 \)(D) \( 3 x^{2}+4 \sqrt{3} x+4=0 \)
- Show that the equation $2(a^2+b^2)x^2+2(a+b)x+1=0$ has no real roots, when a≠b.
- Find whether the following equations have real roots. If real roots exist, find them.\( -2 x^{2}+3 x+2=0 \)
- If the roots of the equation $(b-c)x^2+(c-a)x+(a-b)=0$ are equal, then prove that $2b=a+c$.
- If $a ≠ b ≠ 0$, prove that the points $(a, a^2), (b, b^2), (0, 0)$ are never collinear.
- Choose the correct answer from the given four options in the following questions:Which of the following equations has two distinct real roots?(A) \( 2 x^{2}-3 \sqrt{2} x+\frac{9}{4}=0 \)(B) \( x^{2}+x-5=0 \)(C) \( x^{2}+3 x+2 \sqrt{2}=0 \)(D) \( 5 x^{2}-3 x+1=0 \)
- Determine the nature of the roots of the following quadratic equations: $2(a^2+b^2)x^2+2(a+b)x+1=0$
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies