Question

Please answer this question


the curve y=ax^2+bx+c passes through the points (1, 8), (0, 5) and (3, 20). Find the values of a, b and c and hence the equation of the curve

Answer 1

The curve $y=ax^{2}+bx+c$ passes through the points $(1, 8), (0, 5)\ and\ (3, 20)$.

The co-ordinates given above will satisfy the equation:

$y=ax^{2}+bx+c ...... (1)$

First co-ordinate (1,8):

Putting $x = 1\ and\ y = 8$

$8 = a \times 1^{2} + b \times 1 + c\\\Rightarrow 8 = a + b + c ...... (2)$

Second co-ordinate (0,5):

Putting $x =0\ and\ y = 5$

$5 = a \times 0^{2} + b \times 0 + c\\\Rightarrow c = 5 ...... (3)$

Putting $c = 5$ in equation (2):

$\Rightarrow a + b = 3 ...... (4)$

Third co-ordinate (3, 20):

Putting $x =3\ and\ y = 20$

$20 = a \times 3^{2} + b \times 3 + c$

Putting value $c = 5$

$\Rightarrow 20 = 9a + 3b + 5\\\Rightarrow 3a + b = 5...... (5)$

Using elimination method in equation (4) and (5)

Subtracting equation (4) from equation (5):

$\Rightarrow 2a = 2\\\Rightarrow a = 1$

From equation (4):

$\Rightarrow b = 2$

Hence $a = 1, b =2\ and\ c = 5$

The equation of the curve is:

$\Rightarrow y = x^2 + 2x + 5$