# SAT quadratics: forms, roots, vertex, and solution counts Source: https://1600.now/blog/sat-quadratics-guide --- SAT Math Choose the right quadratic form, solve exact roots, interpret the vertex, and use the discriminant and Desmos without losing restrictions. Written by [Luke Finigan](https://1600.now/about) 4 min read Updated Oct 2, 2026 Use factored form for roots, vertex form for the turning point, and standard form for coefficients and the y-intercept. Solve the equation or graph the function according to the requested quantity, then check any restrictions. ### Know what each form reveals A quadratic function can appear in several equivalent forms. Rewriting is useful when it exposes the information the question asks for. It is not necessary to expand every expression immediately. | Form | Information visible | Example | | --- | --- | --- | | $y=ax^2+bx+c$ | $c$ is the y-intercept; $a$ controls opening | $y=x^2-6x+8$ | | $y=a(x-h)^2+k$ | Vertex is $(h,k)$ | $y=(x-3)^2-1$ | | $y=a(x-r_1)(x-r_2)$ | Roots are $r_1$ and $r_2$ | $y=(x-2)(x-4)$ | ### Connect roots, intercepts, and the vertex The three examples in the table describe the same parabola. Its roots are 2 and 4, so it crosses the horizontal axis at $(2,0)$ and $(4,0)$. Its vertex is $(3,-1)$, midway between the roots horizontally. At $x=0$, it has value 8, giving the vertical intercept $(0,8)$. The sign in vertex form needs care. In $y=2(x+5)^2-7$, the vertex is $(-5,-7)$ because $x+5=x-(-5)$. The coefficient 2 makes the parabola open upward, so the vertex gives a minimum value of $-7$. The vertex's $x$-coordinate and the minimum value are different answers. If the question asks when a modeled quantity is minimized, it may want $x=-5$. If it asks for the minimum quantity, it wants $y=-7$, subject to the model's domain. ### Solve roots exactly when you can Set the function equal to zero to find roots. For $x^2-6x+8=0$, find two numbers whose product is 8 and sum is $-6$: $-2$ and $-4$. Then $(x-2)(x-4)=0$, so $x=2$ or $x=4$. The zero-product rule applies when the product equals zero. You cannot set each factor to zero in $(x-2)(x-4)=3$. First move 3 to the left and simplify, or solve the original equation with another valid method. If factoring is awkward, use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. For $2x^2+x-3=0$, the discriminant is $1-4(2)(-3)=25$. The roots are $\frac{-1+5}{4}=1$ and $\frac{-1-5}{4}=-\frac{3}{2}$. Check them in the original equation. ### Count real solutions with the discriminant For $ax^2+bx+c=0$ with $a\ne0$, the discriminant is $D=b^2-4ac$. If $D>0$, there are two distinct real solutions. If $D=0$, there is one distinct real solution, a repeated root. If $D<0$, there are no real solutions. For $x^2-4x+k=0$, one distinct real solution requires $16-4k=0$, so $k=4$. Two distinct real solutions require $k<4$; no real solutions require $k>4$. If the coefficient of $x^2$ contains a parameter, check whether it can be zero. At that value the equation may become linear, so the quadratic discriminant rule no longer settles the question. Also read whether the prompt asks for real solutions or another condition. Nonlinear Equations and Systems · Easy $\sqrt{w} + 18 = 30$ What is the solution to the given equation? A $4$ B $6$ C $24$ D $144$ - [Open the question and explanation](https://1600.now/bank/math/6b582db1) - [Practice nonlinear equations and systems questions](https://1600.now/bank/math/skill/Nonlinear%20equations%20in%20one%20variable%20and%20systems%20of%20equations%20in%20two%20variables) - [Print the nonlinear equations and systems worksheet](https://1600.now/sat-nonlinear-equations-and-systems-worksheet) ### Use completing the square and root relationships To rewrite $x^2+6x+5$, add and subtract 9: $x^2+6x+9-9+5=(x+3)^2-4$. This makes the vertex $(-3,-4)$ visible. With a leading coefficient other than 1, factor that coefficient from the quadratic and linear terms before completing the square. For roots $r_1$ and $r_2$ of $ax^2+bx+c=0$, their sum is $r_1+r_2=-\frac{b}{a}$ and product is $r_1r_2=\frac{c}{a}$. A question asking for the sum of roots may not require solving either root. For $3x^2-12x+7=0$, the root sum is 4 and product is $\frac{7}{3}$. These relationships come from expanding the factored form. They are useful only after the equation is written with zero on one side and the coefficients correctly identified. ### Graph the right comparison and keep the domain In Desmos, graph the quadratic and inspect roots or the vertex. If the equation is $x^2-6x+8=5$, graph the parabola and $y=5$, or subtract 5 and find the new expression's zeros. The original parabola's x-intercepts do not solve that equation. A decimal display may need an exact algebraic check. Adjust the window before concluding that a parabola has no intersection. In a height model, reject a negative time if the question restricts time to values after launch. In an area problem, a negative length is not a usable solution even if it satisfies the algebra. Practice quadratics in the [SAT question bank](https://1600.now/bank). For each problem, name the requested feature before selecting a method. Use [Desmos practice](https://1600.now/blog/how-to-use-desmos-on-sat) to verify graphs, and the [SAT math formulas guide](https://1600.now/blog/sat-math-formulas) for a broader review. A correct quadratic calculation still needs the right root, coordinate, or value for the question. Try it yourself: For $y=-2(x-4)^2+9$, what is the maximum value of $y$? · A -2 B 4 C 9 D 18 ### Sources - [College Board: SAT content domains and Advanced Math](https://satsuite.collegeboard.org/practice/content-domains) Luke Finigan is a student developer and the creator of 1600.now, a free Digital SAT practice platform. [About Luke](https://1600.now/about). ### Keep reading - **[SAT Math formulas: reference sheet and what to learn](https://1600.now/blog/sat-math-formulas)** Compare Bluebook reference formulas with algebra, percent, circle, and trig formulas to learn, then practice choosing the right one. - **[How to use Desmos on the SAT: equations, tables, and checks](https://1600.now/blog/how-to-use-desmos-on-sat)** Use the SAT version of Desmos to solve equations, inspect intersections, evaluate functions, and check models without relying on screenshots. [Browse SAT resources →](https://1600.now/sat-resources)