Standard ↔ Vertex Form
Algebra, Quadratics
Intro: We complete the square (or expand) with each algebra step shown.
Worked example
- Convert $y = 2x^2 - 8x + 3$ to vertex form.
- Goal: Rewrite $y=2x^2-8x+3$ in $a(x-h)^2+k$ by completing the square.
- Factor out the leading coefficient from the $x$-terms: $$y = 2\big(x^2 - 4x\big) + 3.$$
- Complete the square inside the brackets. For $x^2-4x$, take half of $-4$ (which is $-2$) and square it: $(-2)^2=4$.
- Add and subtract this square inside the brackets so the value of $y$ is unchanged: $$y = 2\Big[\big(x^2 - 4x + 4\big) - 4\Big] + 3.$$
- Recognize the perfect square and distribute $2$: $$y = 2(x-2)^2 - 2\cdot 4 + 3 = 2(x-2)^2 - 8 + 3.$$
- Combine constants: $$y = 2(x-2)^2 - 5.$$
- Read the vertex parameters: $$a=2,\quad h=2,\quad k=-5.$$
- Conclusion (vertex form and vertex): $$\boxed{y = 2(x-2)^2 - 5},\quad \boxed{\text{vertex }(h,k)=(2,-5)}.$$
- Extras (optional): Axis of symmetry $x=h=2$; since $a>0$, the parabola opens upward and has a minimum at $y=-5$.
- Convert $y = -3(x+1)^2 + 7$ to standard form $ax^2+bx+c$.
- Goal: Expand vertex form $y=-3(x+1)^2+7$ to $ax^2+bx+c$.
- Expand the square: $$(x+1)^2 = x^2 + 2x + 1.$$
- Multiply by $-3$: $$-3(x+1)^2 = -3(x^2 + 2x + 1) = -3x^2 - 6x - 3.$$
- Add the constant $+7$: $$y = (-3x^2 - 6x - 3) + 7 = -3x^2 - 6x + 4.$$
- Identify the standard form coefficients: $$a=-3,\quad b=-6,\quad c=4.$$
- Conclusion: $$\boxed{y = -3x^2 - 6x + 4}.$$
- Check (optional by completing the square back): $$y=-3\big[x^2+2x+1\big]+7=-3(x+1)^2+7,$$ which matches the original.
FAQs
How do I get h and k directly from $ax^2+bx+c?$
Use $h = -b/(2a)$ and $k = f(h) = a h^2 + b h + c$. Then $y = a(x - h)^2 + k$.
What if a < 0?
Same process—factor a, complete the square. If $a < 0$, the parabola opens downward and has a maximum at $(h,k)$.
Can I keep fractions exact?
Yes. When $h = -b/(2a)$ is not an integer, keep fractional values for exact forms.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.