I have straight away copied and pasted a worked example from john birds higher engineering mathematics {page 80 problem 4} which reads as follows.
Use an algebraic method of successive approximations to determine the value of the negative root of the quadratic equation: $4x^2 −6x −7=0$ correct to 3 significant figures. Check the value of the root by using the quadratic formula.
The solution goes like this.
A first estimate of the values of the roots is made by using the functional notation method $f(x) = 4x2 − 6x − 7$,
$f(0) = 4(0)2 − 6(0) − 7 = −7$
$f(−1) = 4(−1)2 − 6(−1) − 7 = 3$
These results showthat the negative root lies between 0 and −1, since the value of $f(x)$ changes sign between $f(0)$ and $f(−1)$ (see Section 9.1). The procedure given above for the root lying between 0 and −1 is followed.
What i'am not able to understand is this.
**First approximation
(a) Let a first approximation be such that it divides the interval 0 to −1 in the ratio of −7 to 3, i.e. let $x_{1}=−0.7$.**
What do they mean when they say "Let the first approx be such that it divides the interval 0 to -1 in the ratio of -7 to 3"?