Question

EXERCISE 43
Find the roots of the following quadratic equations, if they exist, by the method
completing the square:
(1) 2x^2-7x +3=0​

Answer 1

Answer:

Step-by-step explanation:

2x2 – 7x + 3 = 0

⇒ 2x2 – 7x = - 3

On dividing both sides of the equation by 2, we get

⇒ x2 – 7x/2  = -3/2

⇒ x2 – 2 × x ×  7/4 = -3/2

On adding (7/4)2 to both sides of equation, we get

⇒ (x)2 - 2 × x × 7/4 + (7/4)2 = (7/4)2 - 3/2

⇒ (x - 7/4)2 = 49/16 - 3/2

⇒ (x - 7/4)2 = 25/16

⇒ (x - 7/4) = ± 5/4

⇒ x = 7/4 ± 5/4

⇒ x = 7/4 + 5/4 or x = 7/4 - 5/4

⇒ x = 12/4 or x = 2/4

⇒ x = 3 or 1/2

Answer 2

Step-by-step explanation:

Given:-

2x² - 7x + 3 = 0

Solution:-

All Equation of the form of ax² + bx + c can should using the quadratic formula:

$\boxed{ \rm \mathrm{\frac{ - b± \sqrt{ {b}^{2} - 4ac} }{2a} }}$

The quadratic formula gives two solution one when ± is adding and one when it is subtracting.

2x² - 7x + 3 = 0

This equation is in standard form ax + bx + c is = 0. Substituting 2 for a, -7 for b, and 3 for c in the quadratic formula,

$\boxed{ \rm \mathrm{\frac{ - b± \sqrt{ {b}^{2} - 4ac} }{2a} }}$

$\rm \: x = \mathrm{\frac{ -( - 7)± \sqrt{( { - 7})^{2} - 4 \times 2 \times 3} }{2 \times 2} }$

Square -7.

$\rm \: x = \mathrm{\frac{ -( - 7)± \sqrt{49 - 4 \times 2 \times 3} }{2 \times 2} }$

Multiply -4 times 2.

$\rm \: x = \mathrm{\frac{ -( - 7)± \sqrt{49 - 8 \times 3} }{2 \times 2} }$

$\rm \: x = \mathrm{\frac{ -( - 7)± \sqrt{49 - 24} }{2 \times 2} }$

Add 49 to -24.

$\rm \: x = \mathrm{\frac{ -( - 7)± \sqrt{25} }{2 \times 2} }$

Take the square root of 25.

$\rm \: x = \mathrm{\frac{ -( - 7)± 5 }{2 \times 2} }$

The opposite of -7 is 7.

$\rm \: x = \mathrm{\frac{ 7± 5 }{2 \times 2} }$

Multiply the denominator 2 times 2.

$\rm \: x = \mathrm{\frac{ 7± 5 }{4} }$

Now, solve the equation $\rm \: x = \mathrm{\frac{ 7± 5 }{4} }$ when ± is plus. Add 7 to 5.

$\rm \: x = \frac{12}{4}$

Divide 12 by 4.

$\rm \: x = 4$

Now, solve the equation $\rm \: x = \mathrm{\frac{ 7± 5 }{4} }$ when ± is minus. And subtract 5 from 7.

$\rm \: x = \frac{2}{4}$

Reduce the fraction 2/4 to lowest by extracting and cancelling out 2.

The equation is now solved

x = 3, x = 1/2

Answer:-

x = 1/2 = 0.5

x = 3