Question
Solve the following quadratic equation: $4 x^2-4 a^2 x+\left(a^4-b^4\right)=0$
✓
Answer
$4 x^2-4 a^2 x+\left(a^4-b^4\right)=0 \Rightarrow 4 x^2-2\left(a^2+b^2\right) x-2\left(a^2-b^2\right) x+\left(a^4-b^4\right)=0$ $\Rightarrow 2 x\left[2 x-\left(a^2+b^2\right)\right]-\left(a^2-b^2\right)\left[2 x+\left(a^2+b^2\right)\right]=0$
$\Rightarrow\left[2 x -\left( a ^2+ b ^2\right)\right]\left[2 x -\left( a ^2- b ^2\right)\right]=0$
$\Rightarrow 2 x -\left( a ^2+ b ^2\right)=0$ or $2 x -\left( a ^2- b ^2\right)=0$
$\Rightarrow x =\frac{ a ^2+ b ^2}{2}$ or $x =\frac{ a ^2- b ^2}{2}$
$\Rightarrow\left[2 x -\left( a ^2+ b ^2\right)\right]\left[2 x -\left( a ^2- b ^2\right)\right]=0$
$\Rightarrow 2 x -\left( a ^2+ b ^2\right)=0$ or $2 x -\left( a ^2- b ^2\right)=0$
$\Rightarrow x =\frac{ a ^2+ b ^2}{2}$ or $x =\frac{ a ^2- b ^2}{2}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
An aeroplane is flying at a height of 300m above the ground. Flying at this height the angles of depression on from the two aeoplane of two points on both banks of a river in opposite directions are 45° and 60° respectively. Find the width of the river. $\big[\text{Use}\sqrt{3}=1.732\big]$
→The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the coordinates of the fourth vertex.
Draw a 'less than' ogive for the following frequency distribution:
| Classes: | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
| Frequency: | 7 | 14 | 13 | 12 | 20 | 11 | 15 | 8 |
A reservoir in the form of the frustum of a right circular cone contains $44 \times 10^7$ litres of water which fills it completely. The radii of the bottom and top of the reservoir are 50 metres and 100 metres respectively. Find the depth of water and the lateral surface area of the reservoir.
→In the given figure, AB is a chord of length 16cm of a circle of radius 10cm. The tangents at A and B intersect at a point P. Find the length of PA.
For what values of k are points A(8, 1), B(3, -2k) and C(k, -5) collinear.
Solve the following quadratic equations by factorization:
$(\text{x}-5)(\text{x}-6)=\frac{25}{(24)^2}$
→$(\text{x}-5)(\text{x}-6)=\frac{25}{(24)^2}$
Show taht the following points are collinear:A(8, 1), B(3, -4) and C(2, -5)
A sum of Rs. 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs. 20 less than its preceding prize, find the value of each prize.
In the following figure, there are three semicircles, A, B and C having diameter 3cm each, and another semicircle E having a circle D with diameter 4.5cm are shown. Calculate:
The area of the shaded region.
The cost of painting the shaded region at the rate of 25 paise per $cm^2$ , to the nearest rupee.
→The area of the shaded region.
The cost of painting the shaded region at the rate of 25 paise per $cm^2$ , to the nearest rupee.