Question 879623
{{{y=x^2-a(x+1)+3}}}

{{{y=x^2-ax-a+3}}}, you should c.... you can use the general solution to a quadratic equation.  Watch for defined variables in the present equation and not confuse them with the instructionally typical variables.


{{{y=x^2-ax+(3-a)}}}


Fitting the parts into the formula places,
roots are {{{x=(a+- sqrt(a^2-4(3-a)))/2}}}
{{{x=(a+- sqrt(a^2-12+4a))/2}}}


Because you want x to be only a single value as a root, you want the discriminant, {{{a^2+4a-12=0}}}.   FACTORABLE:
{{{(a-2)(a+6)=0}}}
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<b>a=2 OR a=-6</b>
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CONCEPTUAL DISCUSSION


"y intersects the x axis", means y=0, so that the solutions for x are the "roots" of y.  


When we solve this for the equation that you were given, {{{y=x^2-a(x+1)+3=0}}}, the application of the general solution of a quadratic formula gives us that x is this formula or expression:


{{{highlight_green(x=(a+- sqrt(a^2-12+4a))/2)}}}.
This or these are the x values for which y crosses or touches the x-axis.
If you specify that y has exactly ONE root, then y intersects the x-axis at ONE point only.  This can occur only for {{{sqrt(a^2-12+a)=0}}}, from the solution of x.  The expression {{{a^2-12+a}}} from your solutions of x is the discriminant.


The typical model for instruction in discussing a quadratic formula is much expressed as {{{Ax^2+Bx+C=0}}}, but typically the lower-case coefficients and constant are used.  The discriminant for this general quadratic equation would be {{{B^2-4*A*C}}}.  In keeping with the known and derivable generality about roots of the equation, the discriminant is zero if there is only one root (or only one x-intercept.)