Solving linear and quadratic equations.
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Linear equations
general form: \(ax + b
= 0\) with \(a\neq 0\)
associated function: line (\(f(x) := ax + b\))
solutions: exactly one: \(x=- \, \frac{b}{a}\) where the line
intersects the x-axis
Quadratic equations
general form: \(ax^2 +
bx + c = 0\) with \(a \neq
0\)
associated function: parabola
solutions: none, one or two
(intersections of the parabola with the x-axis)
There are several techniques to solve quadratic equations.
The equation \(x^2 + px + q = 0\)
has roots \[x_{\pm} = - \frac{p}{2} \pm
\sqrt{\frac{p^2}{4} - q}.\]
Depending on whether the radicand is negative, null or positive,
there are no, one or two solutions, respectively.
Solving quadratic
equations: completing the square
Idea: Use binomial formula to transform \(a x^{2} +bx + c =0\) into \(a(x-x_0)^2+y_0=0\). This yields \[x_{1,2} = x_0 \pm \sqrt{\frac{-y_0}{a}}.\]
Depending on whether the radicand is negative, null or positive, there
are no, one or two solutions, respectively.
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