Question

solve the following quadratic equations by completing the square method 5x2=4x+7​

Answer 1

$\large\underline{\sf{Solution-}}$

Given quadratic equation is

$\rm :\longmapsto\: {5x}^{2} = 4x + 7$

can be rewritten as

$\rm :\longmapsto\: {5x}^{2} - 4x = 7$

On multiply by 5, on both sides, we get

$\rm :\longmapsto\: {25x}^{2} - 20x = 35$

can be rewritten as

$\rm :\longmapsto\: {(5x)}^{2} - 2 \times 5x \times 2 = 35$

On adding 2² on both sides, we get

$\rm :\longmapsto\: {(5x)}^{2} - 2 \times 5x \times 2 + {2}^{2} = 35 + {2}^{2}$

We know,

$\boxed{ \tt{ \: {x}^{2} - 2xy + {y}^{2} = {(x - y)}^{2} \: }}$

So, using this identity, we get

$\rm :\longmapsto\: {(5x - 2)}^{2} = 35 + 4$

$\rm :\longmapsto\: {(5x - 2)}^{2} = 39$

$\rm :\longmapsto\:5x - 2 = \pm \: \sqrt{39}$

$\rm :\longmapsto\:5x = 2\pm \: \sqrt{39}$

$\bf\implies \:x = \dfrac{2 \: \pm \: \sqrt{39} }{5}$

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Explore More :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

• Discriminant, D = b² - 4ac